Choosing an Amp
The main things to consider when choosing an amp are price, your core tone desired, versatility, reliability, and power. Most amps are designed for one or two types of tones and have a tonal signature that is difficult to find elsewhere.
The modern amplifier generally has 2 to 4 channels that you can switch between. This will give you that many distinct tones. Some amps have completely independent controls for each channel, while some amps share some controls across channels which may make them less versatile. Beyond that, some amps have controls that affect all the channels.
Contents
Pre-Amp vs. Power Amp
An amplifier generally consists of two main pieces: the pre-amp and power amp. The pre-amp is generally where the tone-shaping takes place. Modern amps also feature almost all of their distortion here, where it has more control. It also allows you to separate the amount of distortion you can dial in from the amount of volume. Older amps relied on power amp distortion, which meant you had to push the amp to full volume to start getting distortion.
That said, the power section isn't completely transparent, and its tone will vary from amp to amp depending on the design, even when it isn't pushed into distortion.
Overview of Common Channels
Clean
Clean channels are designed to get a relatively clean (undistorted) tone. That being said, some amps' clean channel is still a bit dirty. Sometimes this channel will feature a gain control to dial in pristine cleans vs. mild crunch, but usually not. Even if there is no dedicated gain control, it may still be possible to push the channel into distortion by maxing out the available EQ and channel volume controls.
Simply because tones are clean doesn't mean they'll sound identical across amps. Some amps will have a more compressed sound than others, or have more sag or otherwise alter the attack of the inbound guitar signal. This can get very complex, with the compression affecting some frequencies more than others. Some amps are known to be "warm" while others "sharp" or "cold".
Its also important to pay attention to the tone of the notes as they sustain. Some amps are said to be "organic", while others are "sterile".
Crunch
Crunch is basically a mild to strong distortion, but it usually responds more to attack and playing dynamics than a lead distortion. This channel usually has a dedicated gain control. At lower levels, it will sound like a dirty clean channel. At higher levels, it has a fairly saturated distortion; however, the breakup tends to be a bit noisier or harsher than a lead channel's. In other words, it is less compressed and dirtier than the lead channel, but more compressed and distorted than the clean channel. The distortion also tends to sound more nasal and vintage than modern and tight.
Lead
The lead channel offers you the maximum amount of gain and the most modern tone. The response is tight and the distortion can get very saturated while remaining fairly smooth and harmonic. While it's typically labeled Lead, this channel works well for both leads and distorted rhythm where heavy distortion is preferred over prominent attack. It will almost always have a dedicated gain control to dial in the desired amount of distortion and sometimes has additional tone-shaping controls that other channels don't.
Lead 2
An additional lead channel may simply offer more versatility by being a complete duplicate of the other lead channel or have a different signature to the tone.
Modeling vs. Analog
Traditionally, amps have always been analog electrical devices, focusing both on signature tone-shaping (including adding distortion) as well powering a speaker cabinet. This meant that the amount of tonal control was limited by the complexity of the circuit - the more channels and options, the more expensive the amp would be. Early amps had 1 channel, limited EQ controls, and little or nothing else.
As the cost of electronic components became cheaper and more powerful and computer science developed sound research and methodologies for signal analysis, digital processors began to offer not simply effects, but algorithms that emulated the tone-shaping aspects of amplifiers and mic'ed cabs/speakers. This allowed players to avoid the power aspect of amplifiers altogether, sending their output into DAWs for recording or PAs and monitors for live sound.
Unlike analog amplifiers, modelers could change their tonal processing simply by loading different algorithms, using the same circuit. This allowed them great versatility. Lots of algorithms were made, many designed to emulate specific analog amp models. They could also have their speaker/cab/mic emulation turned off and be connected to the power sections of traditional amplifiers running into traditional guitar speaker cabinets.
The downside of modelers has always been the quality of the algorithms - early units could clearly be differentiated from the analog amps they attempted to emulate, often sounding "fake", failing to replicate certain aspects of analog amps, or displaying artifacts from the digital algorithms such as aliasing or "fizzy" (noisy) frequency ranges. Current generations for the most reputable brands, however, are very difficult to differentiate from their analog counterparts.
Other downsides for modelers are increased complexity and occasional needs for an additional power amplifier. On the other hand, they don't require maintenance or tube replacement like tube amplifiers.
Tube vs. Solid State
Early amplifiers used vacuum tubes exclusively for all their amplification, including in the pre-amp section and the power amp. Later on, transistors (aka solid-state) began to replace tubes for many applications, due to smaller size, cheaper costs, and no necessary replacement or maintenance. However, their application for audio amplification operates somewhat differently than tubes. Many of the things one might find undesirable about tubes from a technical viewpoint are actually desirable to many musicians.
This is particularly true when pushing tubes past their maximum output range, which results in clipping, or distortion. Tubes' distortion is often characterized as "natural", "musical", "organic", and "warm". All that really means is that it sounds pleasing to the ear. Transistors tend to be less pleasing, with a harsher distortion. Tubes also offer aspects, like sag, that cause the technical replication of the amplified signal to differentiate from the source signal over the life of a note. Again, many find this as a positive benefit for musical purposes.
Various technologies have been developed by amplifier manufacturers to get the tube tone out of solid state technology; however, none of them gained enough traction to be considered at the same quality level as tube amps. To this day (mid-2014), the most popular amplifiers rely on tubes for both tone and power. That said, they are also more expensive, require tube replacement and occasional maintenance.
Watts vs. Volume
Amps measure their power in watts, typically 25, 50, 100, 150, or 300 watts. Some new, smaller "lunchbox" amps only have 2, 5, or 10 watts of power. These wide discrepancies may seem like they translate into similar differences in potential volume. However, this is deceiving.
First of all, identical claimed wattage doesn't mean equal volume. Power can be measured by peak power, or by RMS (root mean square), or some other way to define it. RMS is a statistical measure that gives a more accurate picture of what an amp can actually sustain, while peak power only lasts for an instant, which isn't even perceivable. A 50 W and a 100 W amp measured by peak wattage may peak at their respective ratings, but both might have an RMS rating of only 10 watts.
Second, even using RMS power, different designs in amplifiers can exhibit different perceived volume. Tube amps tend to distort differently than solid state amps, which equates to a 100 W RMS tube amp sounding much louder than a 100 W RMS solid state amp.
Third, power does not translate linearly into volume. Double the wattage may only mean 25% more volume, as measured in decibels (+3 relative db). A 20 W amp may sound only half as loud as a 100 W amp. Now, that 25% can make a huge difference in a band context, where you would easily be drowned out. So it does matter, but the numbers don't tell you the complete picture.
This article explains everything pretty well.
A Brief History of Distortion
Early Experimental Distortion Techniques
Early amps weren't designed to produce thick, saturated distortion. Many artists discovered the phenomenon by accident, using amplifiers that were damaged in transport. Some purposefully altered their amplifiers by doing things such as loosening tubes or poking holes in their speakers. Others cranked up their amps as high as they would go, producing a mild overdrive. As the peaks of waveform exceed the amp's maximum output, the peaks would be compressed, or clipped, creating a more square-wave type waveform, which is the equivalent of adding lots of harmonic overtones.
Overdrive
Once artists and manufacturers began to appreciate distorted tones, they found various ways to create them (without irreversibly damaging an amp or speakers). The simplest was to push an amplifier beyond its designed maximum output, creating an overdrive distortion, by running the amplifier at full volume. This could be pushed further by using a high-output humbucking pickup. These were the earliest ways of achieving overdrive distortion. It would later be revisited in the 70's by using solid-state distortion pedals to boost the signal in front the amp, pushing amps much further into overdrive. Amp manufacturers also realized the potential desirability of overdrive distortion, designing their amps to overdrive more easily and with a smoother, more consistent, less noisy distortion.
Fuzz/Dirt/Distortion Pedals
Overdrive was originally reserved for a mild distortion, but artists wanted more. Enter early solid-state distortion pedals: fuzzboxes. These things produced a harsh, noisy, "dirty" distortion but offered artists a much greater degree of distortion. Solid state pedals would evolve more and more, getting away from fuzz and offering more tube-like distortion, in the form of overdrive and distortion pedals.
High-Gain Amps
In the early 80's, amps started to incorporate heavy tube overdrive in the pre-amp section using the smaller pre-amp tubes, with variable gain/drive controls. This allowed real tube overdrive, without having to crank volume to ear-bleeding levels. These started the modern trend of high-gain amps. Various refinements were made, such as cascading gain stages across multiple tubes, offering heavy, saturated distortion without compromising control and minimizing noise and other unwanted warts.
Many artists still run boost pedals in front of high-gain amps, not from any necessity for more distortion than the amp can offer alone, but more due to its impact on frequency response or other tonal/response changes that enhance the amp's distortion. Most amps have distortion voiced for a middle-of-the-road rock tone, while metal artists prefer a tighter, brighter tone that doesn't get muddy on palm mutes. Often using a boost is the easiest way to get there.
Modelers
Most recently, companies are developing digital devices using algorithms to emulate various pedals and amps (including pre-amp and power amp tube distortion). This allowed artists access to a greater range of combinations of pedals and amps and experimentation that would be impossible in real life - things like running amps at full volume with the tubes in class A operation biased as hot as possible or daisy chaining multiple amps.
Overdriving the Amp
Effects Loops
Parallel vs. Serial