What's the purpose of the Father in Fallout 3?
The story is developed throughout the main quest and answers these questions. Taking time to listen to holotapes and talk to NPCs is a great way to get an understanding of what's happening, and has happened, in the game world.
Obviously the rest this answer is just one massive big spoiler, so reader beware!
Character History Before Fallout 3
Essentially, your father in the game was originally working on the foundation of Project Purity - purifying the wasteland's water of radiation. When the player character is born, the mother dies and the father gains entry to a vault in order to bring the child up in an environment with some sort of "normality" (and away from the harsh and unforgiving wasteland). This is also a bonus for your father as he can use the vault's equipment to further his own research.
Father's Reasons
You end up becoming old enough to fend for yourself, and in the end it is apparent that your father has made an important discovery as a result of his in-vault research. He leaves the vault without warning in order to regroup the original project's team and restart the project.
Your Next Steps!
At this point you end up leaving the vault and starting the quest to find your father and sort out what's happening. Along the way you end up becoming a critical part of the project and carry your father's vision - you basically adopt his goal as your own.
Just to expand on James' reasons:
He went to Vault 101 both to protect you (player character) and to find a G.E.C.K. The Vault Overseer used him as a physician and uphold the information that there wasn't a G.E.C.K. there. When James found out about this, he waited until you could take care of yourself to leave Vault 101 and look for a G.E.C.K. elsewhere.
Spoiler
Your Father left the vault to continue his work on project Purity (the project to clean all the water in the area of radiation), but his work was interrupted when his wife got pregnant. The work was very important, though, so he left to continue it.