There are several ways that atoms can combine. In one way, atoms are put together to form things called molecules. To understand molecules, you have to understand what an atom is made up of. Inside the atom, there's things called neutrons, protons, and electrons. The neutrons and protons are all stuck together in the middle of the atom, making up what's called the nucleus. The electrons are much lighter than the protons or neutrons, and they move all around the nucleus. For a picture of this, look at the answer to the question, Picture of an Atom
Some atoms, though, want to have more or less electrons around them than they do. So they form things called bonds that let them share electrons with each other. (There are lots of different kinds of bonds, but this is the basic idea.) When they're sharing electrons, they're totally stuck together and become what's called a molecule.
(added by Mike W.: Those types of bonds are called ionic bonds, where one atom grabs an electron from another and the resulting charged ions stick together by electrical attraction. Bonds can also form even between atoms which are just like each other, so that neither ends up with more or less electrons than it started with. Common examples are the oxygen and nitrogen molecules in the air. Here the reason that the atoms stick together is that the electron clouds, described a bit in that link above, can lower their energy by spreading out over two atoms instead of one, for reasons that one can't begin to describe without quantum mechanics. These covalent bonds are very common and often the strongest bonds around.)
Some atoms, though, don't want to have any more or less electrons than they have. So they don't form bonds with any other atoms. (An example of this is Helium.) So they don't bond at all. (Mike W.: What this really means is that every electron is stuck well, in a low energy state. The next state available would have much higher energy. Only very weak bonds can form with other atoms. The reasons why there are particular states around with different energies are also entirely due to quantum mechanics.))
(Mike W.) The second common way atoms combine is as crystals. They stick together for pretty much the same reasons as atoms in molecules, but instead of making a little unit of fixed size, they just keep stacking up. The copper in a wire, for example, contains many copper atoms stacked up together. A salt crystal contains sodium and chlorine stacked up in an alternating pattern.
Another way that atoms can combine is by something called 'fusion.' This is when the nucleus of one atom and the nucleus of another atom actually come together, making one bigger atom with all of the electrons going around it. This is much less common and very hard for scientists to do. When it happens, it creates a lot of energy, so it would be convenient if people knew how to control it. Unfortunately, as far as we know, in order to make two atoms combine by fusion, you have to use a whole lot of energy (make it very hot), so it's really not a practical way of making energy yet. One place that fusion does happen frequently is in the very center of the sun, where it gets /extremely/ hot.
You should also take a look at the answer to the question Atoms and Molecules
(republished on 07/13/06)