I just want to know what makes a Brahmin. Does that depends on the karma? Does that depends on the fate? Does that depends on learning Vedas?
Some say that next birth of Brahmin is a cow. Is that true?
The qualities of Brahmana are as followed:
BG 18.42 - Steadiness, control, austerity, cleanliness, tolerance, sincerity, knowledge, science and faith are certainly Brahma(na)'s natural Karma
Anyone with predominantly possessing above qualities is Brahmana.
The varna-s are based on natural qualities or karma. Our natural qualities are mostly inherited from our forefathers, hence family lineage (viz caste) also has some role to play, even in today's time. Ultimately in which lineage a person would born is fate/luck. :-)
More details of what various professions may belong to 4 varna-s are described in this answer.
There are many ideas about how one becomes a Brahmana.
Sage Bhrigu's view on Brahmana
The sage Bhrigu said, ‘There is really no distinction between the different orders. The whole world at first consisted of Brahmanas. Created (equal) by Brahman, men have, in consequence of their acts, become distributed into different orders. They that became fond of indulging in desire and enjoying pleasures, possessed of the attributes of severity and wrath, endued with courage, and unmindful of the duties of piety and worship, - these Brahmanas possessing the attributes of passion, - became Kshatriyas. Those Brahmanas again who, without attending to the duties laid down for them, became possessed of both the attributes of Goodness and Passion, and took to the professions of cattle-rearing and agriculture, became Vaisyas. Those Brahmanas again that became fond of untruth and injuring other creatures, possessed of cupidity, - engaged in all kinds of acts for a living, and fallen away from the purity of behaviour, and thus wedded to the attributes of Darkness, became Sudras. Separated by these occupations, Brahmanas, falling away from their order, became members of the other three orders. All the four orders, therefore, have always the right to the performance of all pious duties and of sacrifices. Even thus were the four orders at first created equal by Brahman who ordained for all of them (the observances disclosed in) the words of Brahma (in the Vedas). Through cupidity alone, many fell away, and became possessed by ignorance. The Brahmanas are always devoted to the scripture on Brahma; and mindful of vows and restraints, are capable of grasping the conception of Brahma. Their penances therefore never go for nothing. They amongst them are not Brahmanas that are incapable of understanding that every created thing is Supreme Brahma.
(Mahabharata Santi Parva Section CLXXXVIII )
Sage Bhrigu's view on Brahmana
The sage Bhrigu said, 'He is called a Brahmana in whom are truth, gifts, abstention from injury to others, compassion, shame, benevolence, and penance. He who is engaged in the profession of battle, who studies the Vedas, who makes gifts and takes wealth (from those he protects) is called a Kshatriya. He who earns fame from keep of cattle, who is employed in agriculture and the means of acquiring wealth, who is pure in behavior and attends to the study of Vedas, is called a Vaisya. He who takes pleasure in eating every kind of food, who is engaged in doing every kind of work, who is impure in behavior, who does not study the Vedas and whose conduct is unclean, is said to be a Sudra. If these characteristics be observable in a Sudra, and if they be not found in a Brahmana, then such a Sudra is no Sudra, and such a Brahmana is no Brahmana.'
Mahabharata, Santi Parva, Section CLXXXIX
Yajnavalkya's view on Brahmana
The famous Rishi Yajnavalkya continued,’ ..All orders of men are Brahmanas. All are sprung from Brahma. All men utter Brahma. Aided by an understanding that is derived from and directed to Brahma. I inculcated this science teaching of Prakriti and Purusha. Indeed this whole universe is Brahma. From the mouth of Brahma sprung the Brahmanas; from his arms, sprung the Kshatriyas; from his navel, the Vaisysa; and from his feet, the Sudras. All the orders, (having sprung in this way) should not be regarded as pilfering from one another. Impelled by ignorance, all men meet with death and attain, O King, to birth that is the cause of acts. Divested of knowledge, all orders of men, dragged in terrible ignorance, fall into varied orders of being due to the principles that flow from Prakriti. For this reason, all should, by every means, seek to acquire knowledge. I have told thee that every person is entitled to strive for its acquisition. One that is possessed of knowledge is a Brahmana. Others, (viz., Kshatriyas, and Vaishyas and Sudras) are possessed of knowledge. Hence this science of emancipation is always open to them all.’
(Mahabharata Santi Parva Section CCCXIX)
Yudhishthira's view of who is a Brahmana
Yudhishthira said, " In human society, O mighty and highly intelligent serpent, it is difficult to ascertain one's caste, because of promiscuous intercourse among the four orders. This is my opinion. Men belonging to all orders (promiscuously) begat offspring upon women of all the orders. And of men, speech, sexual intercourse, birth and death are common. And to this the Rishis have borne testimony by using as the beginning of a sacrifice such expressions as -- of what caste server may be, we celebrate the sacrifice. Therefore, those that are wise have asserted that character is the chief essential requisite ... whatsoever now conforms to the rules of pure and virtuous conduct, him have I, ere now, designated as a Brahmana.
Mahabharata Aranya Parva Section CLXXIX
Maheswara's view on who is a Brahmana
Maheshwara said, ‘..Neither birth, nor the purificatory rites, nor learning, nor offspring, can be regarded as grounds for conferring upon one the regenerate status. Verily, the conduct is the only ground. All Brahmanas in this world are Brahmanas in consequence of conduct. A Sudra, if he is established on good conduct, is regarded as possessed of the status of a Brahmana. The status of a Brahma, O auspicious lady [Uma], is equal wherever it exists. Even this is my opinion. He, indeed, is a Brahmana in which the status of Brahma exists – that condition which is bereft of attributes and which has no stain attached to it. of human beings in four orders dependent on birth is only for purposes of classification. The boon giving Brahma, when he created all creatures, himself said that the distribution of human beings in four orders dependent on birth is only for purposes of classification.'
Mahabharata Anusasana Parva Section CXLIII
Sage Pulastya's views on how to become a Brahmana
Sage Pulastya said,"..by vows, by investiture of the sacred, by fasts, by rites, and by Mantras, one becometh a Brahmana."
(Mahabharata, Aranya Parva LXXXIII)
A person need not be a religious person to be a brahmin. There are six orthodox vedic schools and they are defining brahmin in many ways.
The problem is that there are four groups. Vedic hinduism, old vedantians like Adi Shankaracharya, neo-hinduism like arya samaj and agniveer, and neo-vedantians like swami vivekananda.
Vedic hindus and older vedantians, who are now no more, said it is by birth. Arya samaj and agniveer said it is by karma. And neo-vedantians said it is neither by karma nor by birth, it is by swabhava that one is a brahmin.
The concept of brahmin by karma is already vanished. Only islamophobic people follow that because they cannot see a muslim to be called as brahmin. Vedanta no more talks about them. Now it is about birth vs swabhava.
Bhagavad gita says it is by birth and not by swabhava, and it is the same bhagavad gita which says it is not by birth but by swabhava. Same verses different interpretations.
Bhagavad gita chapter 4 verse 13 says,
I have divided the Varnas according to karma and gunas.
The bhagavad gita chapter 14 verse 13 says,
those who dies thinking of any of the three gunas ―sattva, rajas, tamas— in next life they are born in a family whose action is associate with that guna.
Here these two verses are linked.
The older vedantian says that the karma in BG 4.13 refers to the BG 14.13. As karma and guna at the time of death in previous life is the reason of brahmin birth in this life. Though, the family is associate with that action but they not doing it because of their nature of present birth. Note that duties prescribed for brahmins and kshtriyas in BG 18.41 are not actually duties but the universal attributes or behaviour. The faith refers to the faith in himself according to neo-vedanta.
BG 9.32 supports this by saying,
Those of the sinful birth ― The women, the vaishyas, and the sudra― they too get the salvation by surrendering unto me.
BG. 9.33 says,
Then what to say of those pure brahmins and royal sages who meditate upon me and reach me.
The verse 9.32 targets that previous vedic school that says only brahmin and kshtriya get the salvation.
One should also note that according to older vedic schools, the kshtriyas should also learn vedas. A kshtriya, without reading vedas, cannot get salvation even if he dies in battle field.
The neo-vedanta interpret the verse 4.13 for present life. If someone does good karma and have good guna in present life, he is a brahmin. And if someone dies with sattva, in next life he will be born in sattvic family. This good karma or action is driven by gunas / nature / swabhava.
For example, A cow is blocking a road. A sattvic human will takle this problem in a way that it don't hurt anyone including cow.
A rajas swabhava human will hit the cow by hands.
A tamasic swabhawa human will hit the cow with a stick.
About verse 9.32, Neo-vedantians have varying opinions. Some go with Adi Shankaracharya and some have their own interpretation. Neo-vedanta says one need not master sanskrit and vedas to be a brahmin. If one is wise, he is brahmin.
Example of brahmin by birth are: Adi Shankaracharya, Chanakya, Swamy Ramanujacharya.
Example of brahmin by karma: Swami Dayanand Saraswati.
Example of brahmin by swabhava / behaviour / character : swami vivekananda, Albert Einstein, Nikola Tesla, Marie Curie, Ibn al-Haitham, all the scientists, all the intellects, all the wise people of every religion, all the wise people of every field. All these people are brahmins according to neo-vedanta.
This thought of neo-vedanta help the vedanta philosophy to be followed by every part of the world.
Neo-hinduism stops people to eat non-vegetarian food. Only hindu can be the brahmins and that too only by reading vedas. This creates problem as cold countries cannot have vegetarian food every time. This makes the hinduism a religion which can be followed only in specific geography, and not every person, around the world, can read vedas.
Whereas, neo-vedanta says, you don't need to read vedas as the purpose of reading vedas is to become a wise person. If you are a wise person, you are a brahmin. Neo-vedanta allows people to eat anything and that's how it becomes more strong.
That is how neo-vedanta helped the ancient Indian philosophy to be spread around the world.
All the activities that is done by neo-vedanta can be proved by bhagavad gita.
Conduct and character
These are the primary reasons why a brahmin is a brahmin.
He, however, that is born a Brahmana falls away from his status through his own evil acts. Hence, the Brahmana, after having attained to the status of the first order, should always protect it (by his acts).
Neither birth, nor the purificatory rites, nor learning, nor offspring, can be regarded as grounds for conferring upon one the regenerate status. Verily, conduct is the only ground. All Brahmanas in this world are Brahmanas in consequence of conduct. A Sudra, if he is established on good conduct, is regarded as possessed of the status of a Brahmana.
This is my opinion. Men belonging to all orders (promiscuously) beget offspring upon women of all the orders. And of men, speech, sexual intercourse, birth and death are common. And to this the Rishis have borne testimony by using as the beginning of a sacrifice such expressions as--of what caste so ever we may be, we celebrate the sacrifice. Therefore, those that are wise have asserted that character is the chief essential requisite.
I consider thee to be a Brahmana, there is no doubt about this. For the Brahmana who is vain and haughty, who is addicted to vices and wedded to evil and degrading practices, is like a Sudra. On the other hand, I consider a Sudra who is always adorned with these virtues,—righteousness, self-restraint, and truthfulness,—as a Brahmana. A man becomes a Brahmana by his character;
Even a Śūdra who is richly endowed with the knowledge of the Vedas shall become a Brahmin and cultured. Even a Brahmin shall forfeit his Brahminhood and become a Śūdra if his conduct is base and if his diet and culture is base. Brahma himself has said that even a Śūdra should be resorted to like a Brahmin, O gentle lady, if he is virtuous, purified by holy rites or if he has conquered his sense-organs. A Śūdra who clings to his duties, should be considered purer than the twice-born ones. Neither the womb of birth, nor the consecratory rites, neither the Vedic knowledge nor the lineage can be the cause of Brahminhood. Conduct is the real cause. All men are Brahmins if their conduct is pure. Even a Śūdra who strictly adheres to good conduct attains Brahminhood.
If one shows the symptoms of being a brāhmaṇa, kṣatriya, vaiśya or śūdra, as described above, even if he has appeared in a different class, he should be accepted according to those symptoms of classification.
Yudhishthira said, " In human society, O mighty and highly intelligent serpent, it is difficult to ascertain one's caste, because of promiscuous intercourse among the four orders. This is my opinion. Men belonging to all orders (promiscuously) begat offspring upon women of all the orders. And of men, speech, sexual intercourse, birth and death are common. And to this the Rishis have borne testimony by using as the beginning of a sacrifice such expressions as -- of what caste server may be, we celebrate the sacrifice. Therefore, those that are wise have asserted that character is the chief essential requisite.