With so much attention focused on which guitars give you which sound, which guitars look the nicest, and which play the best, the lowly guitar amplifier is often ignored, especially by novice guitarists. Which is a big mistake for guitarists looking to find a great sound. The fact of the matter is; a sub-par guitar played through a great amp can still sound fairly decent, but even the best guitars, when played through a bad amplifier, sound awful.
If you're considering buying your first amplifier, price will obviously be one of your primary concerns. Guitar amplifiers range in price from under $100, to multiple thousands of dollars. A common choice for first amplifiers are the very small and basic 15-watt amps such as the Fender Frontman 15G, which provide a low cost solution to amplifying the guitar. When asked by students and parents, I strongly discourage the purchase of amplifiers like these, for several reasons. First of all, the sound of these amps tends to be tolerable at best, and often sound much, much, worse. Newer guitarists will often become frustrated, complaining that "nothing I ever play sounds good", not fully realizing that it is the equipment, and not what they are playing, that is causing the music to sound inferior. These small amps also don't provide a great deal of volume, which can present a problem. In the beginning, the musical growth of guitar players is quite staggering, and it won't be long before many newer guitarists are ready to start playing with other musicians. Many of these small amplifiers have a hard time being heard above the volume of a drummer, which renders them useless in those situations.
This isn't meant to imply that you need to spend $1000 on your first guitar amplifier. But, by setting your sites above the cheapest, smallest amplifier in the store, you'll certainly end up with an amp that will serve your needs for a much longer period of time. The Fender Pro Junior is a great, low-cost tube amplifier that you'll sometimes even see being used by professional guitarists. What the Pro Junior lacks in control (no EQ, no reverb), it more than makes up for in tone and sound quality (read the full review of the Pro Junior).
There are a few things I'll generally look for in modestly priced amplifiers - at least a 3-band EQ ( low, mid, and high), a clean channel and an "overdrive" channel, reverb, and possibly some sort of "presence" control. Another thing to be aware of is the existence of two types of amplifiers; tube and transistor. The BuddyHawke.com site has provided a relatively simple explanation of the difference between the two. I personally almost universally prefer tube amps, but this is something you'll have to listen to, and decide for yourself. Tube amps are almost always more problematic, and tend to be more expensive.
When shopping for amps, be sure to try many out before you buy one. Play the same guitar through each amp when experimenting in the store. Make sure you spend a good deal of time with each amp; playing them at loud and quiet volumes, with and without overdrive, experimenting with the versatility of sound each amp provides. Do not be afraid to bring your guitar into the store and try it out with the amps you are considering. Try researching specific amps you are interested in on the web. One last thing to be aware of; despite what they may lead you to believe, you can, and should negotiate with music store employees in regards to the price of their merchandise. I have found that with a bit of prompting, I can get at least a 10% discount on guitars and amps not on sale, and often even more.