Shri Krishna is adored in three different ways: vatsalya bhava, madhura bhava and sakhya bhava.
Radha adored Him in madhura bhava. Yashoda and Nanda worshipped Him in vatsalya bhava, and the cowherds with all the sincerity and loving heart adored Him in sakhya bhava, as a friend.
What are the those bhavas described in the scriptures?
As far as I can tell, the terminology of five Bhavas has no basis in Hindu scripture. The earliest reference I can find to it is in a Gaudiya Vaishnava work, Rupa Goswami's Bhakti Rasamrita Sindhu. Here is how Rupa Goswami describes Shanti Bhava:
If the sthāyī-bhāva called śanta-rati mixes with the elements of vibhāva and others that will now be described, and is relished by persons possessing śama [self-restraint], the wise call this śānta-bhakti-rasa. Generally these practitioners possess the happiness of impersonal Brahman, realizing it as the cause of everything. However, such impersonal happiness is dilute, whereas the happiness related to the Lord with form and qualities is intense. The main cause of śānta-rasa arising in such persons is realization of the form of the Lord. The main cause of dāsya and other rasas appearing in the devotees is appreciation of the Lord’s attractive pastimes. Though this is also one factor in the śānta-bhakta’s realization, it is not the main factor. According to the wise, the four-armed form of the Lord (viṣaya) and śānta-bhaktas (aśraya) are the ālambanas [of śānta-rasa].
Here is how he describes Dasya Bhava:
When prīti or adoration achieves a pleasurable nature by vibhāvas, anubhāvas, sāttvika-bhāvas and vyabhicārī-bhāvas in the hearts of the devotees, it is called prīti-bhakti-rasa. When the recipient of mercy acts as a servant, it is called sambhrama-prīti and when the recipient identifies himself as the object of parental affection it is called gaurava-prīti. Those who identify themselves as servants of Kṛṣṇa have sambhrama-prīti for Kṛṣṇa. When this sambhrama-prīti is nourished by vibhāva and other elements it is called sambhrama-prīti-rasa. In sambhrama-prīti-rasa, the ālambanas are the Lord (viṣaya) and His devotees (āśraya).
Here is how he describes Sakhya Bhava:
When the sthāyī-bhāva of sakhya-rati is nourished by suitable vibhāvas and the other elements of the devotee, it is known as preyo-bhakti-rasa (sakhya- rasa). The Lord and His friends of the same age are the ālambanas.
Here is how he describes Vatsala Bhava:
When vatsala-sthāyī-bhāva is nourished by vibhāva and other elements [of bhakti], it is called vatlasa-bhakti-rasa by the wise. The learned say that Kṛṣṇa and the devotees acting as elders are the ālambana for this rasa.
And here is how he describes Madhurya Bhava:
When madhura-rati is nourished by suitable vibhāva and other elements [of rasa] in the devotees’ hearts, it is called madhura-bhakti-rasa. Though this rasa has many components, it will be described in summary because it is difficult to describe, because it is very confidential, and because it is unsuitable for those who do not have a taste for madhura-rasa, since it appears similar to the rasa of mundane love. The ālambanas in this rasa are Kṛṣṇa and His dear, beautiful women.
Some people add to this a sixth Bhava called Kanta Bhava, which involves loving Sriman Narayana as a wife loves her husband, illustrated by Rukmini's love for Krishna, as opposed to Radha's love for Krishna which illustrates Madhurya Bhava. See page 36 of this book.
But even if this classification of Bhakti doesn't have a basis in Hindu scriptures, other similar classifications do. For instance, Prahlada lists nine types of Bhakti in this chapter of the Srimad Bhagavatam:
sravanam kirtanam visnoh smaranam pada-sevanam arcanam vandanam dasyam sakhyam atma-nivedanam iti pumsarpita visnau bhaktis cen nava-laksana kriyeta bhagavaty addha tan manye 'dhitam uttamam
Hearing and chanting about the transcendental holy name, form, qualities, paraphernalia and pastimes of Lord Visnu, remembering them, serving the lotus feet of the Lord, offering the Lord respectful worship with sixteen types of paraphernalia, offering prayers to the Lord, becoming His servant, considering the Lord one's best friend, and surrendering everything unto Him (in other words, serving Him with the body, mind and words) -- these nine processes are accepted as pure devotional service. One who has dedicated his life to the service of Krsna through these nine methods should be understood to be the most learned person, for he has acquired complete knowledge.
And Sutra 82 of the Narada Bhakti Sutras lists 11 types of Bhakti:
guṇa-māhātmyāsakti-rūpāsakti-pūjāsakti-smaraṇāsak ti-dāsyāsakti-sakhyāsakti-vātsalyāsakti-kāntāsakty-ātma-nivedanāsakti-tan-mayāsakti-parama-virahāsakti-rūpai kadhāpy ekādaśadhā bhavati
Although devotional service is one, it becomes manifested in eleven forms of attachment: attachment to the Lord's glorious qualities, to His beauty, to worshiping Him, to remembering Him, to serving Him, to reciprocating with Him as a friend, to caring for Him as a parent, to dealing with Him as a lover, to surrendering one's whole self to Him, to being absorbed in thought of Him, and to experiencing separation from Him. This last is the supreme attachment.
As you can see, these two lists have considerable, though not complete, overlap with Rupa Goswami's list.
Bhakti having emotions in heart is an eternal truth.
Presentation of bhakti in terms of bhavas/rasas is a presentation of Gaudiya Vaishnavism.
On the advice of Chaitanya Mahaprabhu, Rupa Goswami studied Bharat Muni's NatyaSastra with commentaries of AbhinavaGupta etc. and normal hindu sciptures and married Natyasastra and bhakti and wrote a book named Bhakti Rasamrita Sindhu(Nectar of Devotion).
User Keshav Srinivasan has already described what those are in the his answer.
