I should start off by saying that black holes are enigmas. In describing our universe, we have the General Theory of Relativity (GR) to help us describe gravity and objects which are very massive. We also have quantum mechanics (QM) and quantum field theory to help us describe small objects such as particles. For the most part, these two realms are completely separate and these two important theories don't mix. After all, you don't need QM to describe the physics of orbiting the Sun, and you don't need GR to describe electrons orbiting an atomic nucleus.
A problem arises though, when you begin talking about objects which require both QM and GR, and black holes fall very nicely into this regime. Often we find that when we mix the two theories, we wind up with nonsensical answers. It is highly probable that our understanding of the mechanics of a black hole is incomplete. What we really need is a quantum gravity theory to talk about black holes and as yet we don't have one.
That being said, let me lay out the basic structure of a black hole based on what current theory suggests to be true. Fortunately, black holes are relatively simple in structure, as dictated by the No-Hair Theorem (which basically says you only need a few parameters to completely describe the state of a black hole, as opposed to what you'd need to fully describe the physics of a person or even an atom). The (somewhat simplified) anatomy of a black hole is shown below.
In this diagram there are a few important parts. The first being the singularity. When you put a bunch of mass together, gravity acts to squeeze that mass into a ball. That is what drives the fusion in our Sun and makes our planet round. Normally there is some resistive force that stops the matter from collapsing completely, such as the structural integrity of rocks in the Earth or fusion in the Sun. At some point though, if you pack on enough mass, that resistive force can't hold out and gravity wins. In this case, all the matter collapses down as far as it can go. It becomes a point of matter known as a singularity. This singularity has no size, it is literally (and mathematically) a point. All the mass of the black hole exists here. So when they say a black hole is 20 billion solar masses, they mean that the singularity contains the matter equivalent to 20 billion Suns, all confined to that single point.
The other important concept for a black hole is the event horizon. I should stress that the event horizon is not a physical object in space. It is a mathematical boundary. Without getting into the nitty-gritty and in basic terms, the event horizon is where our current physics "stops working". We cannot describe inside the event horizon with our current physics because we are in that region where both QM and GR are important and they don't play well together. The event horizon is also the boundary where, if crossed, you're trapped forever in the black hole. There's a lot more physics involved with the event horizon, but that's the gist. Usually when someone refers to the size of a black hole, they're talking about the event horizon. Thus if they say the black hole is 118 billion kilometers across, they mean that the event horizon has a diameter of 118 billion kilometers. Note, from the image above, that the radius of the event horizon is easily calculated from only the mass of the singularity (and a few physical constants).
There are further components to a black hole such as the ergosphere or photon sphere, and often black holes have accretion disks of material around them. I'll leave those concepts to another question though.