A spectrum is just a fancy term for the different colors of light that are coming from a star. If you've ever shined light through a prism and seen the rainbow of colors that comes out the other end, then you've seen a spectrum. The light that comes to us from stars is very similar - although it looks to our eyes like it is just one color, it is actually made up of many different colors. Astronomers can break up the light coming from a star (by using a technique similar to shining it through a prism) and measure how much light is coming at each color, and they can use this information to figure out information about the star, like its temperature.
Also, the spectra that we see coming from stars often contain what look like dark lines at particular colors, which means there is much less light coming from the star at that color than at the nearby colors. This usually means that the star's atmosphere contains certain types of molecules which absorb light of that color, so we don't see as much of it coming from the star. Astronomers can use the information from these "spectral lines" to figure out what a star is made of.
Here is a nice example of a spectrum of the sun.
Note that in the above I've been talking about visible light - light that we can see with our eyes. But a lot of the light in the universe has a "color" that our eyes aren't able to detect - examples of this light are radio waves, ultraviolet light, x-rays, etc. Even though we can't see this light, it is still being emitted by stars, and astronomers can build instruments which are able to detect it and look at its spectrum as well.
This page updated on June 27, 2015