With this distinction I am attempting to clarify an apparent riddle in Kant's thought, namely, things in themselves, and why we can talk about them despite not knowing them.
An object is encountered by us in a two-fold way: first, an object, so far as it is given to us, is an appearance; second, so far as the object is not given to us, is is a thing in itself. One may wonder: how do we encounter the object so far as it isn't given to us?
Something preliminary to deal with is the term 'object' itself. This term doesn't specifically point out an appearance, but rather our own thought: the object, so far as it appears to us is an object because it is thought by us. This should make it easier to point out that the object so far as it doesn't appear to us is also thought by us.
We know the object so far as it appears because the content given by the appearance can determine the concept of the object synthetically. So far as the object doesn't appear, it can't provide any material for a synthesis, and so remains a thought of an object in general.
The distinction is between an object so far as it is given (appearance), and an object so far as it is not given (thing in itself). The parent of this distinction can be discovered by removing what has been attributed on the positive side of the distinction.
When we remove the 'so far as it is given', we have only 'an object.' Without the appearance we are left with the mere thought of an object in general, and this is the parent concept of appearance and thing in itself.
Any number of examples could be given for appearance - one would merely need to experience the world to discover them. In all of these cases the thing in itself is also thought in relation to the appearance, and no more than just thought.
Another way to bring this home is to point out that appearances are appearances of something. They aren't appearances of themselves, but of the object. However, we only have an object so far as we think one, and we only are given a determinate object so far as it appears. So far as it doesn't appear, it is the thing in itself.
Observe an object from a few angles. So far as you are observing it with some determinate content (color, size, &c) it is an appearance, but it yet remains the same object while the appearance changes. From the permanence of the object in relation to the impermanence of its particular appearance we can see what has been designated the thought of the object in general which has remained the same over the course of the experience. This object, so far as it was just thought by us and not given, is called the thing in itself.
The thing in itself is entirely indistinct, since it is just thought by us. This thing in itself could be anything, it doesn't have to have properties like the ones that it appears to you as, but it is at least unmistakably thought by us in relation to this appearance.
Importance in Kant's Work
Kant's predecessors also characterized objects in general in the study of ontology. This general conception of objects was considered a priori knowledge. Kant, on the other hand, considered the object in general to not be known by us, except so far as it was combined with an intuition of it. The concept of an object in general was only the form of an object (or experience generally), and not knowledge.