The 'open asset import library' has a model viewer where you can preview the models with all aclaimed supported formats (including animations). I would suggest exporting your animations, to a variety of formats and see if any of them give you wrong results (i.e. render artifacts) and conclude your own awnser. Personally, I had the most success with Collada (.dae) although I used 3ds Max.
To support the animation itself (presumable skeletal): interpolate the animation data (aiNodeAnim) between the key-frames; you need to calculate the bone matrices from the interpolated animation data (create a matrix out of position/rotation etc); calculate the 'animated' vertices by transforming the 'vertices' (aiMesh/aiBone) with the bone matrices that influence them; weight all of the 'animated vertices' and sum them together; render the newly vertices. And this a basic high level overview of animations.
There are tutorials and books that can go deeper into the subject. For example ogldev's tutorial 38, Jason Gregory's Game Engine Architechture has dedicated animation chapter. You can take a look at the source code of assimps model viewer.
A game engine is a development framework for developing games. Unity and UDK are game engines. OpenGL and DirectX are rendering engine. A game engine may have its own rendering engine.
Game Engine Architecture by Jason Gregory is a nice book
Take a look at: Game Engine Architecture ISBN-13: 978-1568814131