According to Rig Ved an Apsaraa is a female counterpart of Gandharv. They connote beauty. Their synonyms are "nymph", "celestial nymph", and "celestial maiden". They are supernatural in nature and most of them are dancers in Indra's court. They usually entertain Devtaa and fallen heroes. Indra uses them for his purpose also. Only Apsaraa mentioned in Rig Ved is Urvashee who married to Pururavaa - an ancestor of Paandav and Kaurav. (Rig Ved, Book X, Hymn 95)

A beautiful article is availabale on Internet written by Lokmanya Bal Gangadhar Tilak - "On the Etymology of Urvashi and Apsaras - Extract from Chaldean and Indian Vedas (1917)" - it discusses the connection of Apsaraa, water, and 'thigh' (uru).

In MBH, Stories/1-Dushyant, Shakuntalaa reminds Dushyant, that there are six foremost Apsaraa - Urvashee, Poorvchitti, Sahajanya, Menakaa, Vishwaachee and Ghritaachee. Since Menakaa was born from Brahmaa Jee, she is the first and foremost one. There are many others - Chitralekhaa, Chitrasenaa, Dandgauree, Gopaalee, Janapadee, Kumbhayoni, Madhuraswan, Mishrakeshee, Prajaagaraa, Pramlochaa, Punjikaasthalee, Rambhaa, Saha, Swayamprabhaa, Tilottamaa, Vishwaachee, Varoothinee,  etc.

Naatya Shaastra - the principle work of dramatic theory for Sanskrit drama, lists the names of the following Apsaraa: Arjunee, Budhaa, Devadattaa, Devasena, Dhriti, Kalabhaa, Kerala, Magadhee, Manjukeshee, Manorama, Mishrakeshee, Nandaa, Santati, Saralaa, Saudaaminee, Sudati, Sukeshee, Sulochanaa, Sumalaa, Sumukhee, Sunandaa, Sundaree, Supushkalaa, Supushpamaalaa, Vigagdh, and Vividhaa.

Thus the total number of the names of Apsaraa are as follows (see their Names in Table form) --

Adrikaa (gave birth to Satyavatee), Arjunee,
Budhaa,
Chitralekhaa, Chitrasenaa
Dandgauree, Devadattaa, Devasena, Dhriti,
Gopaalee, Ghritaachee, Hemaa (mother of Mandodaree)
Janapadee
Kalabhaa, Kerala, Kunmbhayoni
Madhuraswan, Magadhee, Manjukeshee, Manorama, Menakaa (Vishwaamitra), Mishrakeshee,
Nandaa,
Poorvchitti, Prajaagaraa, Pramlochaa, Punjikaasthalee (mother of Hanumaan),
Rambhaa
Saha, Sahajanya, Santati, Saralaa, Saudaaminee, Sudati, Sukeshee, Sulochanaa, Sumalaa, Sumukhee, Sunandaa, Sundaree, Supushkalaa, Supushpamaalaa, Swayamprabhaa
Tilottamaa,
Urvashee (married to Pururavaa, curse to Arjun)
Varoothinee, Vigagdh, Vishwaachee, Vividhaa.

The following are some Apsaraa who had the contacts with human beings --

--Urvashee married to Pururavaa and gave birth to six sons.

--Urvashee got attracted to Arjun, and when Arjun did not marry her, she gave him Shaap to be an eunuch for his whole life, then Indra requested her to be lenient and this Shaap was reduced to one year only and whenever he wanted to start it.

--Urvashee was sent to Maharshi Sharadwaan to distract him from his Tapasyaa and this caused the birth of a twins - one boy Kripaa and one girl Kripee. The King Shaantanu brought them up. Kripaachaarya became the Kul Guru of Kaurav family and Kripee was married to Drone.

--According to MBH 1/10, Janapadee Apsaraa was sent to disturb Sharadwaan's Tapasyaa. This resulted in a twins - one boy and one girl - Kripaa and Kripee.

--Menakaa was sent to disturb Vishwaamitra's Tapasyaa and this resulted in the birth of Shakuntalaa who married the King Dushyant, whose son was Bharat - the ancestor of Kaurav and Paandav.

--Pramlochaa Apsaraa was sent to disturb Muni Kandu's Tap, and this caused the birth of a girl named Maarishaa who was married to Prachetaa Gan and gave birth to Daksh.

--Punjikaasthalee Apsaraa was given Shaap to be born on Prithvi as a she-monkey, and this she-monkey was married to Kesaree and gave birth to Hanumaan who is an earnest devotee of Raam.

--According to MBH, Tilottamaa rescued the world from the Raakshas Sund and Upasund.

--Bharadwaaj Muni kept his semen which fell after seeing Ghritaachee Apsaraa and Dronaachaarya was born from it.