To surveil is to spy on someone, the way a television private eye sits in her car and uses binoculars to surveil the show's villain.
The noun surveillance came first, from the French surveiller, "to oversee or watch." If you were to surveil someone, you'd keep a close eye on them, monitoring everything they do. These days, that means keeping track of their online activity as well as their phone calls and physical movement. It's usually authority and governmental figures doing the surveillance — although if a company wants to sell you something, they just might surveil your internet history.