"Objects are simple" (2.02). As states of affairs relate to complex facts, so do Wittgenstein's objects relate to the complex objects we encounter in the world. For instance, a chair is a complex object made up of a seat, a back, and legs. The seat, back, and legs can further be analyzed into simpler parts. Objects are the basic units of this kind of analysis; they cannot be analyzed further into simpler parts.
An object has internal and external properties. The internal properties are its logical form: what kind of object it is and how it can combine with other objects in states of affairs. The external properties are what is true of it, what states of affairs it does occur in.
The internal properties of an object hold remain true no matter what. No matter what is the case, there will always be these objects with these logical forms. Thus, objects and their internal properties are what make up the substance of the world (2.021). We can imagine worlds other than this one, but we cannot imagine worlds that do not have the same logical form as this one. That is, we can imagine a world where horses speak and grass is pink, but we cannot imagine a world without space, time, or color.
Wittgenstein is careful to note, however, that the subsistence of these objects and their internal properties tells us nothing about the material properties of the world (2.0231). The material properties of the world are determined by objects being combined with other objects in certain states of affairs; that is, by the objects' external properties. For instance, the internal properties of yellow and red are indistinguishable: they are both colors and they can both occur in the same sorts of states of affairs. The only way we can distinguish red from yellow is by their external properties, by saying that certain things are true of yellow that are not true of red, and vice versa. Wittgenstein says that "substance is what subsists independently of what is the case" (2.024).
There is no metaphysical "glue" that binds objects together in particular states of affairs: "in a state of affairs objects fit into one another like the links of a chain" (2.03). The metaphor of links of a chain suggests that it is the form of the objects themselves that allows them to be combined in states of affairs. It is how objects link together that determines the structure of the state of affairs, not some properties or relations external to the objects.
Wittgenstein now comes back to where he started. Some states of affairs are the case ("positive facts") and some states of affairs are not the case ("negative facts"). Whether one state of affairs is the case or not has no bearing on whether any other state of affairs is the case or not. The world is the sum total of those states of affairs that are the case.
Wittgenstein never tells us what objects or states of affairs are; he simply tells us that they are the simplest kinds of things and facts there are. We can infer that by "objects" he cannot mean things like tables and chairs, because these can be divided into smaller parts. The difficulty lies in determining what exists that can't somehow be further analyzed. Wittgenstein never gives us an example of an object, remaining maddeningly vague as to what the base units of his ontology might be.
There are three main interpretations of what Wittgenstein takes objects to be. The first interpretation identifies objects as the basic elements of sense data. This interpretation reads Wittgenstein as following Bertrand Russell, who argued that all empirical description could be analyzed down to basic sense data with which we are directly acquainted. There are two problems with this reading of Wittgenstein, however. First, Wittgenstein wants to take objects as things that can combine in basic states of affairs as being either true or false. However, a speck in the visual field can be a whole variety of different colors, which makes it difficult to simply divide states of affairs into the categories of "true and "false." Second, if Wittgenstein did mean sense data when he talked about objects, he probably would have just said so.
The second interpretation argues that objects are nothing quite so specifiable as sense data, but that they do denote a basic, underlying structure to the universe. Objects are the basic building blocks of reality, even if we cannot determine precisely what they are.
The third interpretation denies that objects have any kind of independent being at all. This reading emphasizes Wittgenstein's repeated assertion that objects can only exist within states of affairs and that the world is made up of states of affairs and not of objects. Objects do not exist as the basic building blocks of reality, but rather are given being only in the context of states of affairs.
But why does Wittgenstein assert that there are simple objects at all? Proposition 2.021 says, "Objects make up the substance of the world." "Substance" is a term that was used a great deal in 17th and 18th century rationalist thought. The idea is that the world must have some underlying substance that is unchanging and indestructible. Wittgenstein applies this idea to objects, but in a distorted manner. Objects are unchanging and indestructible, but they are also empty logical forms: we can learn from them nothing about what the world is like until they are combined in states of affairs (2.0231). There is an unchanging, indestructible, and necessary logical form to the universe, but everything that is the case depends on contingent states of affairs.
This logical form determines, for instance, that purple is a color, and that we can only use the word "purple" in contexts where a color word is called for. What kind of a color purple is, what sorts of things in the world are purple, are contingent facts, but the simple fact that purple is a color is a necessary aspect of the logical form of "purple." For reasons we will explore later, Wittgenstein would argue that to say "purple is a color" is plain nonsense, but we will let it stand for now.
If the world had no substance, and if there was no logical form, then "purple is a color" would be a contingent fact, just like "that lampshade is purple." If "purple is a color" were contingent, that means it could be otherwise; it could be that purple is a number. And if purple were a number, then "that lampshade is purple" would be nonsense. Now we grasp the meaning of 2.0211: "if the world had no substance, then whether a proposition had sense would depend on whether another proposition was true."
Wittgenstein never gives us an example of an object because there is nothing to be said about objects. Asking "what is an object?" is like asking "what does everything have in common?" The best answer Wittgenstein can muster to this question is that everything shares in common a logical form that allows it to occur in states of affairs. Objects are the simplest, most general things there are: the only thing that all things hold in common is that we can say something about them.