Olivia Dunham is the main character. She's an FBI agent who gets caught up in this crazy series of science experiments that someone is basically trying out on the world at random. She and her FBI superior (Phillip Broyles, a creepy dude) consult the help of Dr. Walter Bishop, who used to do similar work back before he wound up in a mental facility. (He is by far the most fascinating character. Genius, but still mentally unbalanced and prone to major mood swings. A mad scientist gone soft, if you will.) His son is Peter Bishop (Joshua Jackson, of Dawson's Creek fame), about whom we know very little but the show keeps revealing a little bit more about his past each week.
Just to warn you, the show is REALLY far out. None of the "science" is remotely realistic.