Step 1: What purposes will your amp need to cover? Jazz combo? Metal band? Cover band work? Just practicing at home? Know the purpose will help narrow your choices. At the same time, many amps can cover more than one type of use.
Step 2: With amps you really do get what you pay for. So buy the best one you can within your budget. If you're into really low-fi sounds, you'll be able to get some grit for little money. But reaching the heights on tonal Nirvana costs some bucks. You can always move up later if you need to.
Step 3: Find out what your favorite guitarists are playing through. Chances are you've heard a particular guitarist's sound and thought "That's what I'm looking for!" Most players consistently use the same types of amps. Look for articles and interviews with those guitarists that discuss gear.
Step 4: When you go to a store to try out amps, be sure to use your own guitar. Most stores are cool with this, as different guitars will make the amps sound different. If you're working with a knowledgeable sales person at the store, they may be able to also direct you toward amps you may not have thought of. But don't let them sell you stuff you don't need.
There are also a ton of amps built for other instruments: bass, electronic drums, keyboards, vocal PAs, etc. Ignore all those for now.
Once you've narrowed the field a bit with those four steps, we can look at the different types of amplifiers available.
First consider what's driving the amp:
Tube Guitar Amplifiers
This is the oldest type of amp, but many guitarists still consider them the best sounding.
Pros: Awesome classic tone, power saving, louder than solid state amps at the same wattage
Cons: Tubes are breakable and will need to be changed every so often
Solid-State Guitar Amplifiers
Transistors are used in place of vacuum tubes.
Pros: Much sturdier for heavy use, good for particular guitar sounds like 80's rock
Cons: Harsher tone on the high end and overall thinner tone compared to tubes
Digital Guitar Amps
These are the newest breed of digital modeling amps.
Pros: Very versatile with a variety of sounds and effects including modeling of much more expensive amps. Fantastic for cover bands who need a huge selection of sounds.
Cons: Tones sound fake on cheaper models though more expensive ones sound great. There's a steeper learning curve to learn how to get all the sounds you want.
Hybrid Guitar Amps
These combine tubes and solid-state transistors. Oftentimes with the tubes running the amp and transistors used for effects and such. While the tone won't be dead on like a good tube amp you'll get close for a lot less money.
You can get all of these amplifiers in a selection of sizes and wattages from tiny 10 or 15-watt single 8" speakers, to giant 200-watt full stacks with 4-8 12" speakers. Make sure you go through those first four steps to see what you really need. While that giant full stack is always tempting, you may regret it when you get it in your apartment or try to do a coffee shop gig. For home practice, all you need is a little 15-watt amp. Your neighbors will appreciate it. I use a 5 watt Vox DA5 for practice. It also fits in my suitcase as an emergency amp at gigs. It's not loud enough on its own, but sounds nice when mic'd through a PA system.
If you're working with a band, you can take a look at 1x12 or 2x12 combo amps. Lots of companies make really good ones like Fender, Line 6, Mesa Boogie, Soldano, and others.
If you're moving up to bigger clubs, then you can take a look at half and full stacks. Great tone, great looks. But I guarantee you'll get tired of hauling them around unless you've got roadies.
I keep at least 3 amps on hand. A tiny practice amp like the Vox described above. A Line 6 2x12 modeling amp for studio work and smaller gigs. And a Marshall half stack for much larger shows.
Bottom line: Look at what your needs are and compare them to the guidelines above to see where to start your search. Don't just go by size, wattage, or heaven forbid, looks. Check out what your favorite players are playing and then sit down in a store and try out as many as you can until you find the one you like.
Life's short... Play loud.