I'm surprised there's no "lesson" on purchasing a good amp. I'm certainly no expert, but I want to help people avoid the same mistake I've seen several times and even made myself - buying or just starting with a shitty amp. This is presumably an easy mistake for someone with no real guitar experience or knowledge of what to purchase. Maybe you knew every single tube amp inside-out before you ever picked up a guitar, or maybe all your friends play guitar so obviously it's common knowledge, but in either of those cases this article is not for you. This article assumes you're just going with an amp and not bothering with pedals. I'm also writing this for someone who is just starting guitar... If you already know something in a section, just skip it, rather than complaining that some people weren't born with the knowledge you've learned.
If you're too lazy or ADD to keep reading, I guess the single best point to take from this is: If you don't have an amp, go try different amps, particularly digital modelling amps, at your local music store before settling on one. If you bought a guitar that came with a starter amp, throw the amp away and take your guitar to try out different amps at your local music store, and buy one that sounds good and you like.
So you've bought your first guitar and you're looking for an amp. Or you bought a starter kit that came with a shitty starter amp because it sounded like a good package deal and you didn't know better. Don't sweat it. If you've never played on anything but your starter amp (or never played on anything), you might not realize how much of a difference the amp makes. It can make a huge difference between sounding crappy and sounding good, and moving to a better amp can alleviate a lot of frustration with your sound and playing. If you are planning on playing clean or with light distortion, a cheap starter amp *might* be all you need, but if you're not sure, read on.
If you have a shitty starter amp, chances are it's solid state. Here's a breakdown of the three main types of amplifiers, in case you don't already know about them. You can also just look this all up on Wikipedia or whatever.
Solid-state: Solid state is electric but not digital, using semiconductors or transistors or what have you. These are generally less expensive and easier to repair, but the cheap ones sound like shit. If you bought a guitar starter pack, chances are it came with an awful solid-state amp.
Vacuum-tube: Vacuum tube amps use vacuum tubes (gasp), which arguably produce the best sounding distortion/overdrive by the way they shape a clipping audio signal. Don't worry about the technical stuff so much, or look it up. Chances are if you are just starting, you shouldn't bother with a tube amp - they are more expensive and require more maintainance.
Digital: The newest variety of amplifier, digital amps using digital modelling; in other words, computer-processing on a chip emulates the sound of other (sometimes much more expensive) tube or solid-state amps. The built-in effects are also accomplished digitally. Digital amps are designed to provide a lot of flexibility at a reasonable cost. Some guitar players argue that digital amps sound "too digital" and will bash on them with a vengeance, but I think a decent digital amp gives perfectly acceptable tone and is probably enough to satisfy the average listener. I've never heard a digital amp that really sounded digital. Also, digital digital digital digital maverick.
Technically, an amplifier is just the part that amplifies the signal from the guitar (and adds effects if it has them); what most people think of when they say amp is actually a combo amp, which is an amplifier combined with one or more speakers in a speaker cabinet. On the other hand, amp stacks will have the amp head sitting on one or more separate speaker cabinets.
An important facet of an amp that you should note is the controls it offers. A shitty amp will probably have just a few controls. The most basic are Volume and Tone (tone effecting the sharpness of the sound). Many amps have a gain knob, which effects the amplification (rather than just the volume) of the guitar signal and is where you'll get the crunchy, distorted sound used in rock, metal, etc. However, in a crappy amp, you won't get very good sounding distortion just from cranking the gain all the way, and it might introduce feedback squeal, static, or other unwanted noise. Going on, you might find equalizer controls, such as bass and treble knobs, and effect controls, such as reverb and flanger. Higher-end amps may have dozens of knobs. Many are self-explanatory, but you might want to ask a store employee if you're confused by anything.
If you're just starting out, I highly recommend going with a small and relatively inexpensive digital amp, such as the Roland Micro Cube. The Micro Cube gives you great sound and a lot of flexibility in a tiny and extremely portable package. You'll get remarkably good tone, distortion, and effects out of the Micro cube, probably more than enough to impress casual listeners who aren't obsessive tone-junkies. It also can run on AA batteries, so it'll still do you good when you're a world-class shredder and you can run up and down the block blasting solos at your neighbors. Of course, the Micro Cube isn't your only option; it's success has inspired other companies to release similar "micro"-style modelling amps with big sound and great effects at remarkably affordable prices.
Of course, you shouldn't just take my word on it - again, go to your local music store and try out a variety of amps, and try to find one that best suit's your wants and needs, with the tone, effects, and output that feels right for you. You might be surprised what kind of sound you can get out of a small modelling amp costing less than $150. Make sure you test as much of the amp as you can - try the different amp models, play with the effects and controls, see if you can get what you want out of it. Ask the employees before cranking it up, but try to give it at least a basic volume test - you probably do not want to max the volume and play in a cramped indoor environment, but try to get a feel for how loud it can get.
There will always be rich spoiled brats on Ultimate Guitar and elsewhere who will endlessly harass and ridicule you because your little practice digital amp you bought for learning how to play doesn't cost as much as the $2000 Marshall stack they got from mommy and daddy for their half-birthday, but don't take their comments to heart. An inexpensive digital amp is perfect for the beginning guitarist, giving you tone that won't frustrate you in a package that isn't a huge investment.
I hope this article will be useful to some who are just beginning to play electric guitar, or even some who have been playing for some time on a crappy amp and didn't realize it was holding them back. I appreciate comments and feedback, but try not to be a dick. Not everyone is rich, not everyone is born an expert on amps or has friends who have been playing guitar for years, and as of the time I am writing this, there are zero lessons on amplifiers.