I have heard this term Dwija a lot in this site but I am not sure why are some brahmins called Dwija or 'Twice Born'. I know not all but only some special ones are called Dwijas but how are they different from the other ones. Do we have other similar terms also for Thrice born or N times born ?
Not only Brahmins, Kshatriyas and Vaishyas are also called twice born.
Manu Smriti 10.4. Brahmana, the Kshatriya, and the Vaisya castes (varna) are the twice-born ones, but the fourth, the Sudra, has one birth only; there is no fifth (caste).
The first birth is when they are born (naturally) of their mothers' wombs and the second when they receive the initiation of the Gayatri mantra (this ceremony in which they receive the initiation is known as the Upanayana or the sacred thread ceremony).
Initiation is regarded as a new birth and hence the second birth.
2.147. Let him consider that (he received) a (mere animal) existence, when his parents begat him through mutual affection, and when he was born from the womb (of his mother).
2.148. But that birth which a teacher acquainted with the whole Veda, in accordance with the law, procures for him through the Savitri, is real, exempt from age and death.
In the above quote, 147 is talking about the first birth of a twice-born and 148 about the second.
In the second birth, the Guru (teacher who initiates) is regarded as the father and the Mantra is considered as the mother.
´There are four classes: Brahmin, Kshatriya, Vaishya Sudra. Three classes are twice-born: Brahmin, Kshatriya and Vaishya. Of these --
The first birth is from the mother and the second at the tying of the Muñja-grass girdle. At the latter birth, the Sa¯vitrı verse is his mother, while the teacher is said to be his father.
Vashishta Smriti 2.5.2-1-3