How to make an animation / armature remember poses? I rigged a character and I can render stills, but I can't make an animation. How to make Blender "know" that I want the arm - as an example - go from up to down?

I tried to move my character arms down, than changed the current frame to 1 and moved it up a little. Than I went to frame 2 and moved it up a bit more and repeated this until frame 20. When I went back to frame 0, it stayed as I left it in frame 20, so I guess I'm not doing it right.

How to "remember" the poses?
You need to set keyframes after posing the character. With the posed object selected hit I and set the appropriate keyframe, e.g LocRot (for location and rotation). You need to do this before you move onto the next frame.

Frames that have keyframes on them are highlighted with an orange line in the timeline. If you don't see the keyframe before moving on, your changes will be lost for that frame and you will have to redo it.



You may want to enable Auto Keyframing (Docs) to automatically insert keyframes as you pose your armature: