Atoms are made of a nucleus orbited by electrons. You'll hear people say that the nucleus and electrons are much tinier than an atom itself, so that most of the atom is actually empty. This might seem weird because the Empire State Building is made of atoms and it certainly doesn't seem like "mostly empty space." I'll try to explain what's going on with an analogy. In the process, I'll explain what we mean by the "size" of the nucleus.
Let's start with a slightly different question: Is the solar system full of stuff, or is it mostly empty space?
Suppose we were giant galaxy-sized beings, too clumsy to stick our hands into the solar system and feel around. How would we answer this question? Well, physicists have learned that a good way to probe something is to throw things at it and watch what comes out. So let's throw a baseball at the solar system at, say, 60mph and see what happens. This baseball is moving incredibly slowly compared to all the planets (which are going more like 60,000mph), so it will start to fall in towards the sun. It might hit the sun, or it might sling around the sun and come flying out in a different direction. Either way, every time we throw in a baseball, pretty much no matter how the baseball starts out, it will get deflected (and maybe sucked in) by the sun's gravity.
To compute how big the solar system is, we should throw lots of baseballs at it at random angles and see how likely it is that they'll be deflected. It's very likely, so we'd conclude that the solar system appears HUGE to a 60mph baseball, and we'd be right.
But now suppose instead of a baseball, we send in a spaceship going at 60,000,000mph. Most likely, that spaceship will just zoom right through the solar system without encountering anything. On rare occasions, the unfortunate ship might smash into the sun, or a planet, or an asteroid. We galaxy-beings would count up the number of times this happened and conclude that the solar system looked relatively tiny to a 60,000,000mph spaceship.
An atom is the same. In physics, we define the size of a particle in terms of the likelihood that it will deflect a probe (usually another particle) that we throw at it. As in the case of the baseball vs. the spaceship, the answer depends on how fast the probe is going. In everyday life, things are moving pretty slowly compared to the speed electrons orbit the nucleus. Thus, atoms look big and "full of stuff" at these speeds. The Empire State Building is made from a bunch of atoms that are moving very slowly with respect to each other.
But if we instead use high energy probes, we'll find that they usually just zoom right through the atom (like a spaceship). On rare occasions, they might be deflected by the nucleus. To probes which are fast enough to fly through the electron cloud, but not too fast to sail through the nucleus, an atom looks 100,000 times smaller. This is the famous Geiger–Marsden experiment
(usually called the Rutherford experiment).