We represent facts to ourselves by means of pictures. The elements of a picture correspond to the elements of a fact, i.e. the objects that constitute it. If three objects combine in a particular way to form a fact, then the picture of that fact will consist of three elements combined in a similar way. Wittgenstein calls this combination of elements in the picture the "structure" of the picture and he calls the possibility of this structure "pictorial form" (2.15). That is, that a picture is the kind of thing that can arrange its elements in a certain determinate way is due to its pictorial form.
A picture must have something in common with what it represents in order to depict it properly (2.161). A painting must exist in space if it is to depict things that exist in space, and it must have color if it is to depict colors (2.171). Similarly, a picture of a fact must have a "logico-pictorial form" in common with that fact in order to depict it. Though a fact is made up of objects and a picture is made up of pictorial elements, they are both structured in the same way due to this common form.
Just as a spatial picture represents things in physical space, a logical picture represents things in logical space. A logical picture represents possible states of affairs: it is the most general kind of picture because logical form is the most general kind of form. However, a logical picture cannot represent logical space or logical form itself, in the same way that a spatial picture cannot represent physical space itself. Rather, it displays its form by depicting facts (2.172).
Logical pictures represent possible situations, which we can then compare with reality. The situation represented by a picture is the sense of the picture (2.221). If this sense agrees with reality (if what the picture depicts is the case), the picture is true. If not, the picture is false. We cannot tell just by looking at a picture whether it is true or false: we must compare it with reality (2.223).
"A logical picture of facts is a thought" (3). That is, a thought is a logical picture of a possible situation. Because thoughts must share the logical form of what they are about, it is impossible to have an illogical thought. Expressing an illogical thought is as impossible as representing a geometrical figure that contradicts the laws of space (3.032).
We express thoughts by means of propositions (3.1). Propositions are communicated by means of propositional signs through modes such as speech, writing, or body language. Like a picture, a proposition represents a possible state of affairs by sharing a form in common with it; i.e. its elements are arranged in a similar way. A random string of words cannot have a sense because there is no internal coherence in the way that these words are arranged. This is the upshot of 3.1432: "Instead of, 'the complex sign "aRb" says that a stands to b in the relation R,' we ought to put, 'That "a" stands to "b" in a certain relation says that aRb.'" A proposition does not say what relation holds between its elements; rather, that relation is what makes the proposition sayable.
Up to 2.1, the Tractatus was dealing with ontology, i.e. what there is. At 2.1, Wittgenstein shifts ground from discussing what there is to discussing how it is that we can make sense of, and communicate, what there is. He shifts from questions of ontology to questions of language, thought, and representation.
One of the most famous ideas of the Tractatus is that propositions are logical pictures of facts. Wittgenstein's use of "picture" is semi-technical, being somewhat literal and somewhat metaphorical. He is not giving "picture" a different meaning from its ordinary usage so much as he is expanding that usage. In saying, "We picture facts to ourselves" (2.1), Wittgenstein is saying that conceiving of something is a matter of picturing it to ourselves. If something can be the case, we can conceive of it, and that means we can make a logical picture of it.
There is a direct correspondence between logical pictures and facts: for every fact, there only one logical picture that corresponds to it. We can tell which fact a logical picture depicts, because the picture shares the same logical form as the fact.
Wittgenstein illustrates this point at 2.1512 and 2.15121 with the example of a ruler laid against an object to measure its length. The ruler and the object have nothing in common except that they both have length. But because of this commonality, we are able to relate the one to the other. There only needs to be one point of contact to relate two very different objects to one another. Both a ruler and a measured object have length, and so it is possible to relate aspects of the object to different graduated lines and numbers inscribed upon the ruler. Similarly, both a logical picture and a fact have logical form, so it is possible to relate elements of the fact to elements in the logical picture.
When, at 2.172, Wittgenstein says that a picture cannot depict its pictorial form, he is making the important distinction between saying and showing. Though a picture may have the same logical form as a fact, it cannot depict this logical form. Rather, the logical form shows itself in the picture. The significance of this sharp distinction between what can be said (facts) and what can be shown (form) will become clear later on.
In discussing thoughts, Wittgenstein is not making any psychological claims. Throughout the Tractatus he keeps well away from both psychology and epistemology: he is interested in how things are, not in how we perceive things to be. In discussing thoughts, he is saying only that thoughts must share a logical form with propositions and with reality in order to reflect them. He is not talking about the content of thoughts—how they work, where they come from, etc.—he is only talking about the form of thoughts. In doing so, he is saying only that they must adhere to the same logical form that everything else does. When he denies the possibility of illogical thought at 3.03, he is not saying we cannot think things that are contradictory (e.g. "It is raining and it is not raining"), but rather that we cannot think things that have no sense. I cannot think, "the number two is purple," because it is not even clear what that thought would be.
Propositions are thoughts in a communicable form. Propositions, thoughts, logical pictures, and facts all share a common logical form. We know that a certain proposition expresses a certain thought, and that that thought is a logical picture of a certain fact, because proposition, thought, picture, and fact all have the same logical form.
However, propositions are not the same thing as sentences. We might draw the distinction by saying that a sentence is what we utter and a proposition is what is said. A sentence is what Wittgenstein calls a "propositional sign": it is a written (or spoken) manifestation of a proposition.