I'm using Bullet in my game (
http://bulletphysics.org). It was well received and respected from its very first release. It has been implemented in Blender, some notable games, and even a few feature films. The API is fairly simple, but the documentation is (IMO) not up to par. Given that, it took me the better part of a weekend to implement it into my game.
Oh ... and yes, it does have a jump feature (check out btKinematicCharacterController).
Some other physics APIs are nVidia's PhysX, Open Dynamics Engine (ODE), and Newton (sorry I don't have links readily available ... google will find them).