Whether you're using software or animating free-hand, there are some basic concepts that apply. Knowledge of basic physics, especially the Newtonian laws of motion, will help make your characters move and interact in believable ways. It's important to know how objects will behave when they bump into each other, or when forces such as gravity and friction act upon them. The attributes of the objects themselves (think size and mass) will also affect how they should move and interact in your animated world.
One commonly cited animation principle is that many actions in nature follow an arc, including animal and human motions. The study of how body parts or other grouped objects and their joints move to get from one place to another is called kinematics, and it's a term you'll hear often in relation to animation and animation software. To some extent, animators learn these rules so that they can creatively break them. Rather than striving for entirely realistic action, they often exaggerate movement, at least a little, to keep things lively.
Animators use a number of other methods to make interesting animations with fluid motion. For instance, rather than have things move at constant speed or start and stop abruptly (which is rarely the way things move in the real world), the animator will incorporate slow-in and slow-out, meaning he or she will make them start slowly and accelerate to a faster speed, or decelerate and slow down before coming to a complete stop. Follow-through and overlapping can also be used to create more realistic and interesting motion. Flexible objects, or objects attached to other objects, will not move all at once. One part might start the motion and the other is pulled along, arriving at its final position (or follow through) a bit later than the first part. For instance, the upper part of a leg moves, followed by the calf, then the foot, with hair or clothing lagging behind and catching up after a person moves. There's also a lot of natural overlap to action. Rather than having a character complete one action before starting the next (which would likely look unnatural and boring), he can begin the next action before the last ends.
Two other concepts often applied to non-rigid objects in animation are squash and stretch, which are flattening and lengthening of the object, respectively. For instance, a ball might squash a bit when it hits the ground, or a balloon might stretch a bit when pulled by its string. Another, anticipation, is used to broadcast an upcoming motion by moving an object or character in one direction just a little before it makes a larger motion in the other direction.
Animators have always had to plan out the timing of actions to build suspense or to make sure things happen at the right pace. They also have to take scene composition into account (including contrast, lighting, perspective and what objects are visible) to build the right mood, convey the intended story and make sure that the audience can tell what's going on.
A lot of old-school animation was done by having the main artists draw out key frames that showed the beginnings and ends of motions, and having more junior artists draw the more numerous in-between frames that fill in the action from the first key frame to the next.
Another technique used in traditional animation was layering using see-through animation cels. Artists would create background images that could be reused, and then they'd draw the foreground elements and characters on animation cels that could be layered over the backgrounds and photographed. This means the background wouldn't have to be drawn in every time.
These, along with many other basic animation concepts and methods, are still used by people animating on physical media and on software, and in some cases they have been worked into the basic functionality of the software packages.