There are three Kashyap.

(1) A traditional doctor was also named Kashyap who was going to cure Pareekshit when he was destined to be bitten by Takshak, but Takshak sent him back by bribing him lots of money. Read his story here - MBH, G-0-Prolog/10.

(2) Another Kashyap was a Braahman who was also in Mahaabhaarat, Anu Geetaa-8, Anu Geetaa-10, Anu Geetaa-12. He was engaged in long discussion with a Siddh.

(3) The third Kashyap is the son of Maharshi Mareechi. Maharshi Mareechi is one of the ten Maanas Putra (brainchildren) of Brahmaa Jee - Mareechi, Atri, Angiraa, Pulastya, Pulah, Kratu, Bhrigu, Vashishth, Daksh and Naarad. Mareechi was married to one of the 9 daughters of Kardam Rishi - Kalaa. They had a son named Maharshi Kashyap. Kashyap Jee was married to 13 daughters of Prajaapati Daksh. Their names were - (1) Aditi, (2) Diti,  (3) Danu, (4) Vinataa, (5) Kadroo, (6) Taamraa, (7) Khasaa, (8) Arishtaa, (9) Iraa, (10) Surabhi, (11) Krodhvashaa, (12) Surasaa, and (13) Muni.

Once Kashyap and Aditi did a lot of Tap to get a son. Pleased their Tap when Bhagavaan appeared before them, He asked them to ask for a boon. Both asked for a son like Him three times. So Bhagavaan appeared in their house three times as their son - (1) once as Vaaman Avataar, (2) another time as Raam to Dasharath and Kaushalyaa, and (3) the third time as Krishn to Vasudev and Devakee.

Kashyap's Family
(1) Diti had two sons - Hiranyaaksh and Hiranyakashyap and a daughter Sinhikaa who was married to Viprachitti - Danu's son. When Vishnu killed her two sons - Hiranyaaksh and Hiranyakashyap by His two incarnations of Varaah and Narasinh respectively, she got very bereaved. She served Kashyap Jee very well and asked him for a Var that she should have a son that can kill Indra. Kashyap Jee told her to observe a special fast to get such a son - Punsavan Vrat.

She kept the fast diligently but once she made a mistake and as Indra was looking for an opportunity to destroy that child, he entered her stomach and cut her embryo into seven pieces. But that embryo did not die, rather it was divided in seven children. Seeing this Indra again cut each piece in seven. Still those seven pieces did not die rather they now became 49. Then they said to Indra - "Why are you killing us, we are your brothers?" So Indra stopped cutting them. Later they joined Indra and they became 50.

When Diti woke up, she found that her 49 sons were with Indra. She got very happy to see them and asked Indra - "I wanted a son who could kill you, but how this son became 49. Tell me if you know it." Indra also told her everything that he was looking for an opportunity to destroy her child and as he got that he cut the child in seven pieces and then again the each piece in seven pieces. But because of the effect of her Vrat, they did not die and all the 49 pieces remained alive. They had joined him now and asked for her forgiveness. These 49 sons of Diti were called Marut Gan.

(2) Aditi had 12 Aaditya. Later she had Vaaman Avataar in her house. Naarad Puraan, Poorv/6 says that "when Vaaman Avataar appeared, he told Aditi, "I have been born as your son in your two previous lives. Now also I will give you joy being your son."

But surprisingly we have not come across this information anywhere else and who were those two Avataar He appeared two times as their son - Raam and Krishn were after Vaaman Avataar.

(3) Danu gave birth to many great Daanav like Andhakaasur, Viprachitti, Vrishparvaa, Raahu etc.
(4) Vinataa was the mother of Arun (Soorya's thighless charioteer) and Garud (Vishnu's ride).
(5) Kadroo gave birth to many great snakes.
(6) Taamraa gave birth to six daughters;
(7) Khasaa gave birth to Apsaraa, Yaksh, Muni etc;
(8) Arishtaa gave birth to Gandharv;
(9) Iraa gave birth to trees and creepers;
(10) Surabhi gave birth to cows;
(11) Krodhvashaa gave birth Krodhvash Gan; and
(12) Surasaa gave birth to snakes.
(13) Muni gave birth to
Thus the whole Prithvi was filled with Kashyap Jee's children.

Other Stories
A story comes in Skand Puraan, in Shiv Rahasya, that there was an Asur named Akhirsen who had a daughter named Surasaa. Because of Asur's defeat at the hands of Devtaa, Daitya Guru Shukraachaarya taught her many kinds of Maayaa and she became well versed in Maayaa. She thought that if she had children from some great Rishi, her children will be more powerful than Devtaa, so she approached Kashyap Jee who, at that time was busy in Tap. When he opened up his eyes he found a very beautiful girl in front of him. He yielded to her proposals. Both had happy time for long time and during this time Surasaa gave birth to three sons and one daughter - the first son was Surapadm, the second was Sinhmukh and the third one was Taarak - read his story at Taarakaasur. The daughter’s name was Ajaamukhee. Of the sons, Sinhmukh had a lion’s face and Taarakaasur had an elephant’s face; and Ajaamukhee had the face of a goat.

Where the Kashyap Name Appears
Kashyap name has been mentioned in
Rig Ved (9.114.2),
Raamaayan - Baal Kaand - (70, 75) - this Kashyap is the same as of the son of Mareechi
Mahaabhaarat - Aadi Parv (63) - this Kashyap was a Braahman who wanted to get some money after reviving Pareekshit from Takshak bite
Vishnu Puraan - (1.15 and 21) - this Kashyap is the same as of the son of Mareechi
Bhaagavat Puraan - (4.1) - this Kashyap is the same as of the son of Mareechi