Martha is a Caltech science or humanities graduate student/employee who seemed to be very interested in Sheldon. She is good friends with Abby and appeared in the episode "The Psychic Vortex" (S03E12).
As Leonard and Howard are both off on a double date with Penny and Bernadette, Raj feels lonely, and suggests to Sheldon that they go out and have some fun. After much persuasion and actually giving away his Green Lantern lantern, Sheldon agrees to go with Raj. At the University Mixer, they meet Abby and Martha, where Martha instantly became interested in Sheldon's limited edition Green Lantern lantern and power ring. All four of them head back to Sheldon's apartment, where they are seen playing Rock Band (presumably).
Raj yet again is interested in going out with Abby, but she will only agree to hang out with him if Martha can come along. Raj bribes Sheldon with his Incredible Hulk hands signed by Stan Lee. Martha and Sheldon engage in conversation, where she reveals her interest in Edwin Abbott's novella Flatland. As Sheldon states it's getting late and he has to go to bed, it is seen Martha suspects that she has to go with him by saying "Okay.". Sheldon shocks her by saying: "Goodnight puny human!".
As Abby and Raj are getting busy in the living room, Martha makes her escape to Sheldon's bedroom, asking if she could hang out there for a while probably hoping for coitus. Sheldon accepts actually letting someone into his room, and immediately leaves saying "I'll go sleep in Leonard's room" surprising his visitor. Martha has not been seen since.