Dr. Siebert is the president of the California Institute of Technology. Judging from his acts in the show, he seems to be responsible for getting funding for the university, and supervises how the funding are used.
In "The Monopolar Expedition" (S2E23), He sent Sheldon on a three-month National Science Foundation expedition to the Arctic Circle, and was annoyed when Sheldon visited him at his home in the middle of the night. His wife eventually set their dogs on the unwelcome visitor. Siebert does not appear on the screen.
In "The Vengeance Formulation" (S3E09), When the President visited Barry Kripke's lab to inspect a micro-controlled plasma experiment and review how a grant was used, he (along with the board of directors) became a victim of Sheldon's "foamy vengeance" on Kripke, in which large quantities of a solution of hydrogen peroxide and one of saturated potassium iodide were secured above the ceiling tiles of the lab's drop ceiling and triggered to react with ordinary dish soap, creating an exothermic release of oxygen.
Two years later, In "The Benefactor Factor" (S4E15), Siebert organized a get-together with a number of the university's wealthy donors in order to garner funds for the university's programs, and ordered Leonard, Sheldon, Howard, and Raj to attend, calling the guys his favorite geniuses. Seeing Sheldon's behavior towards the donors, however, he asks that he not appear again. He states the board of directors insists Sheldon has a beautiful mind, but Siebert thinks Dr. Cooper "is just bananas." He also encourages Leonard to have sex with a rich female donor in order to get funding.
In "The Vacation Solution" (S5E16), President Siebert forces Sheldon to take vacation. It is also revealed that Sheldon often gives him suggestions at that time - first talk with the President while they're urinating in the men's room, and then installs suggestion boxes everywhere without asking the President and put written suggestions in them - and the President likes neither. Siebert also reveals that the most-often received suggestions in these boxes is "Can Dr. Cooper take a vacation?"
In "The Rothman Disintegration" (S5E17), when Sheldon and Kripke were fighting over who would get Rothman's office they asked Siebert, who was urinating, to decide who gets the office he said he had to let them work it out, as he couldn't care less who inherits said office.
In "The Parking Spot Escalation" (S6E09), when Sheldon is claiming on a parking lot space against Howard, he tries to visit Siebert's home again. and this time the President does not even answer the door. Sheldon later calls him at home, and it can be deduced from Sheldon's talking that the President is rather annoyed, and even throws "salty language" in the call. Siebert does not appear on the screen.
In "The Discovery Dissipation" (S7E10), Siebert gets angry again because of Sheldon's inappropriate behavior when the latter received a radio interview of NPR. He tells Sheldon that the university needs funding, and Sheldon suggests to him that he put out a swear jar to raise money. It is implied Siebert then uses swear words on Sheldon, for Sheldon then tells him that that statement would be worth a dollar.
After talking with Amy and Wil Wheaton, Sheldon is back on the phone with the university president telling him that he is ready to do more interviews and for him to be enthusiastic about his discovery. The president thanks him, while Sheldon tells him that he should thank his friend Wil Wheaton. Sheldon says he has nine friends, but the president doesn't want to be his tenth.
Siebert does not appear on the screen in this episode.