Free Physics Engines

NVIDIA PhysX SDK (Windows, Linux)
NVIDIA's PhysX engine is free for both commercial and non-commercial use on Windows and Linux systems. The Xbox 360 and PlayStation 3 (PS3) support as well as source code are not free. Its rigid body object physics system features realistic simulation of objects taking into account things like momentum, acceleration, velocity, position, rotational motion, friction, collisions, constraints, impulse, reference frames, etc. It has collision primitives like the sphere, box, capsule, plane, heightfield, convex shape and triangular mesh. It also has a variety of joint types, ragdoll creation and editing, and so on. Other features include advanced character control (auto-stepping, dominance groups), ray-cast and articulated vehicle dynamics (with wheel shapes and joint-based suspension), support for multi-threading, the Ageia PhysX Accelerator, volumetric fluid simulation (SPH or smooth particle hydrodynamics, implicit meshes, two-way interaction with rigid bodies, cloth or soft bodies), cloth and clothing authoring, etc.
FarSeer Physics Engine (Windows, Xbox 360)
This is a 2D physics engine for Microsoft's XNA and Silverlight platforms. It features collisions (with concave and convex polygons support, multiple collision geometries per body, line-line, line-AABB, line-Geometry collision methods, etc), joints (revolute, angle, slider, pin), force controllers (linear and angular spring, fluid drag and buoyancy, ability to build custom controllers for things like explosions, steering, etc), and so on. It is released under the Microsoft Permissive License (Ms-PL).
Open Dynamics Engine (Windows, Mac OS X, Linux, etc)
This open source C/C++ library for simulating rigid body dynamics features advanced joint types (ball and socket hinges, sliders, fixed, angular motor, etc), integrated collision detection (with collision primitives like the sphere, box, capped cylinder, plane, ray and triangular mesh) with friction support, etc, and can be used for simulating virtual creatures, vehicles and other objects in virtual reality environments. It is released under the BSD licence. The library is cross-platform can be compiled for Windows, Mac OS X, Linux and probably other systems. On Unix type systems like Linux and Mac OS X, you will need an X11 server.
Bullet (Windows, PS3, Xbox 360, Wii)
This is an open source 3D collision detection and rigid body dynamics library that is free for commercial use. It uses the COLLADA physics file format. Platforms supported include Windows, PlayStation 3, XBox 360 and Nintendo Wii. It is released under the Zlib licence.
APE - Actionscript Physics Engine (Flash)
This is a free Actionscript 3 physics engine that you can use in Flash and Flex. It features circle, rectangle and wheel particles, spring constraints, collisions, etc. The engine is released under the MIT licence.
Havok Physics and Animation (Windows)
This SDK includes libraries (in binary form only), samples and documentation for programmers creating games and other programs for Windows. The last time I checked, it's free for non-commercial games and applications. Please read their licence for details.
Newton Game Dynamics (Windows, Mac OS X, Linux)
This is an engine for real time simulation of physics environments. It supports scene management, collision detection, etc. There are versions for Windows, Mac and Linux. You are required to notify the developers of any product that uses the engine.