Atoms have a nucleus ( composed of protons and neutrons) and a cloud of electrons. The number of the different components varies between different elements.
Electrons can only have specific energy levels in an atom - that is the energy is not continuous but quantized. This is different from, say, the solar system, because a planet could theoretically be in any orbit. But we know from quantum mechanics (and many, many experiments) that electrons can only occupy certain "orbits."
Sometimes an electron can be bumped into a higher orbit (the higher orbits have less energy, because they are not as tightly bound) or kicked out of the atom altogether. If a second electron moves into the spot it will release energy as it falls into that level.
Hydrogen is the easiest element to study. The nucleus is a single proton, and it can have only one electron. The different energy levels can be illustrated like this:
(though in reality electrons don't travel on nice orbits)
If an electron in the second lowest level does a "quantum leap" to the very lowest, it will emit energy - light - at a very specific frequency. In this case 656 nanometers, which is red. If, on the other hand, an electron falls from the 3rd level down to the 1st level the emitted light is blue.
A "spectrum" will show all of these distinct levels, with the brighter ones being the most common. This shows the spectrum of hydrogen with a diagram of the quantum leaps.
In other atoms it becomes more complex, not every jump is allowed, and in the presense of a magnetic field a "fine structure" is seen in the spectrum, as some of the main lines look double.
The result of all of this is that each different atom will give a different characteristic spectrum. So, you can use this to identify different elements in anything from crime scene samples to distant galaxies.