Perhaps a dumb question...

What exactly does a phono preamp do? Remove noise? Clear up dynamics? What is the purpose? I've noticed even my school's relatively cheap listening rooms uses a very basic phono preamp, I'm just interested in what exactly the effect of it is that makes it worth spending money?
Two purposes ...

Raise signal level so that appropriate volume at the speakers can be reached ...

Equalize the signal off the LP which is not recorded flat, but rather to the standards of an RIAA Equalization Curve.

Good Listening!
Phono preamps do primarily two things.

First, the signal generated by the phono cartridge is quite small. It is limited by what the needle can do when it vibrates in the record groove and moves a tiny coil in a tiny magnet (or a magnet insided a coil). This is a much smaller signal than what comes out of a CD player, a tuner, or a tape machine, all of which have the benefit of built-in amplification stages. The phono preamp amplifies the signal from the cartridge so it is about as big as the others. Accordingly, the line level preamp receives a signal that is usably loud.

Secondly, the phono preamp reverses the RIAA volume curve treatment. An interesting truth about the physics of putting sound directly into a record groove is the low notes require huge waves in the vinyl and the high notes are represented by really small waves. This would be hard for the cartridge and needle to handle. As a result, in the early days of records, the RIAA (Recording Industry Association of America) set a standard where the low notes would be put on records at reduced volume, to make the waves smaller, and the high notes would be put on records at increased volume, to make the waves bigger. This would allow the needles and cartridges of the day to satisfactorily capture both highs and lows. When records are played back, this has to be reversed. The phono preamp amplifies the low notes and attenuates the high notes so that the music sounds like it should.

Depending on the phono preamp, a particular unit might also do things like adjust for proper capacitance loading for the cartridge.

As you can see, the phono preamp is not optional, if you have a record player. Sometimes phono preamps are built into preamps, and you don't need a separate box. This was more true in the old days when records were the primary way people played music at home. Nowadays, record players are relatively rare (sigh), and many manufacturers don't bother to build phono preamp stages into their preamps. So we have to buy stand alone units if we want to play records.
Here's a simple explanation. A phono stage is needed to boost the relatively low voltage signal (about 0.3 - 3.0 mV) from a phono cartridge up to what is considered a line-level signal (about 100 - 200 mV). I don't remember all of the numbers but it takes about 40 to 70 dBs of gain, depending whether the cartridge is MM or MC, to boost the signal from a phono cartridge to a resonable level that most people can listen to confortably. So the phono stage is a pre-amplifier. Most pre-amps these days do not have a built-in phono stage and even those that do have one do not always have a very good one. The remedy is to use an external phono stage.
To reproduce the sound on an LP you must use either a built-in or stand alone phono preamp. Basically a phono pre-amp does two things. One, it converts the information embedded in the groove according to something called the RIAA (Recording Industry Association of America) curve. The curve is used during the cutting of LP's to compress the signal at its frequency extremes. Two, it amplifies the very low level signal produced by the phono cartridge. A cartridge's stylus riding in the groove generates a very faint signal. If you put your ear close to a record while it's playing you can hear it.

Back when turntables were the only game in town all stereo preamps, integrated amps and receivers had built-in phono pre-amps. When the popularity and availability of LP records waned manufacturers dropped the phono sections from their products. So nowadays systems with turntables use stand alone phono pre-amps.
I have had phono preamp for years but couldn't explain its function. The above explanations are very helpful, thanks audiogoners. There is no dumb question.
But if my receiver or amp has an input that says "Phono" as well as a knob that says "Phono output" or something like that, is it safe to say that it has a built in phono stage?
You better check your owners manual before you assume there is a phono stage built into your receiver. But anyway you'll know for sure when you plug it in. With no preamp you'll get crap out...
We cannot assume that your receiver has a built in phono stage. Some amps just label their inputs for convenience. As Sugarbrie said, the best thing to do is to check the owner's manual. If you can't find the owner's manual, just plug your turntable into the input labeled phono. If nice music comes out, then there is a phono stage. If funny sounding (too low in volume or too tinny with little bass) music comes out, then there is no phono stage. Can you tell us which receiver you are using? Maybe someone has a similar one and can tell you whether it has a phono stage.
Pioneer VSX-9900. I'm pretty sure it's got a phono stage, it doesn't sound TOO crappy when I play records through it.
To completely reiterate what everyone else said, here's a little reading for you, at your leisure ;)