I'll bite, and will probably get bitten, but below are some of the technical considerations off-the-cuff. Sorry for the length! This is all on basic power amplifiers; preamps and integrated amplifiers add features to the mix that also influence their desirability for a given application.
1. Amplifier output impedance. As covered in some of the technical threads, this affects how the amplifier interacts with the speaker. Speaker impedance (ohms) changes over frequency, sometimes widely (or wildly) in the case of many conventional speakers, sometimes fairly smoothly but with a fairly wide range such as ESLs that drop in impedance as frequency increases, and some like Magnepans and other planer-dynamic designs do not vary much at all.
An amplifier's output (driving) impedance also changes over frequency, typically increasing as frequency goes up and devices lose gain and feedback goes down. I have some examples in various threads here and over at the ASR site (the latter articles have been updated a time or two with more modern amplifiers and speakers).
SS amplifiers usually have output impedance much, much (orders of magnitude) lower than tube amplifiers and so tube amplifiers will interact more with the speaker. A SS amp is closer to an ideal voltage source, which IME is the goal for an audio amplifier (yes, that is not always true). If the speaker's impedance does not vary much, this doesn't really matter. If the speaker does vary significantly over frequency, then the frequency response you hear will be affected by the amplifier you choose. Then it becomes preference; you choose an amplifier that works with the speakers to produce the sound you like to hear.
Higher amplifier output impedance can also lead to more distortion from the speaker, but for me the speakers themselves usually dominate the over distortion from the system, so that is minor compared to the frequency response changes.
Edit: Damping factor is 1/output impedance, so a higher damping factor means a lower output impedance. I have read anything over 20 is enough for many speakers; I tend to look for a damping factor of 100+ but again it depends upon the speakers. My Maggies were happy with tubes and a damping factor of <10; my Salon2's are more demanding.
2. Noise floor, or signal-to-noise ratio (SNR). If you have highly-sensitive speakers like horns then you may be more sensitive to noise from the amplifier (and anything up to it, of course). Hiss and hum that are inaudible on some speakers may be intolerable on another. For example, the hiss may be inaudible on an 80 dB/W/m speaker, but move to a 100 dB/W/m speaker and the noise is four times as loud and may be rather annoying.
3. Power, of course, as you need enough power to play cleanly at the volumes you listen. This actually involves SNR as well because SNR is normally specified relative to full power. A 10 W amplifier with 100 dB SNR referenced to max output has a lower noise floor than a 100 W amp with the same 100 dB SNR relative to full power. If you need a lot of power, you may need to look at the SNR as well.
As for the actual power required, IME most folk overestimate their average power and underestimate their peak power needs. This is very dependent upon the speakers, listening distance, and listeners' preferences, of course. But usually a watt or two is all that is needed for average listening, but 100 W or more is needed for dynamic peaks if you want to avoid clipping. (Note clipping on a very loud passage, such as a cymbal crash or explosion, probably does not really matter since the sound is already loud and brash.)
I usually listen around 70 dB or so average level and I have a pair of fairly low-sensitivity speakers at 84 dB/W/m. I sit around 8 feet away and need just 0.1 W from the pair to reach 70 dB at my ears. Only 1 W produces 80 dB, which to me is a loud average level. If my speakers were say 94 dB/W/m, I'd only need one-tenth the power for the same volume.
For peaks, various studies have put dynamic music at around 17 dB peak-to-average level, a power factor of 50. Movies even higher, 20 dB or more, a factor of 100 in power. That means if I crank it up and use 1 W to each speaker to produce 80 dB average at my ears, I need about 100 W if I do not want to clip the peaks.
The other factor at play is how we hear, according to things like the Fletcher-Munson equal-loudness curves. These show that we are less sensitive at very low and very high frequencies. It may take 20 dB (100x power) or more greater level in the deep bass (100 Hz and below) to match the loudness of the sound in the midrange. Now that big 100+ W amp doesn't look so big after all if it is driving full-range speakers. This is also one reason for using subwoofers since you may be able to get more power in their limited bandwidth and also use more of them to both boost the volume and counter room modes.
4. Technology. The old tubes vs. solid-state (SS) trade, or A vs. AB vs. G/H vs. D amplifier classes. For now I'll skip the amplifier class; a good design should sound good and have good specs irrespective of the class of operation.
Tube amps usually have higher output impedance, dissipate more power, and have higher distortion than SS designs. They tend to be single-ended and thus have larger even-order harmonic distortion which I have often read is more benign to listeners. It is interesting that this is by design; intrinsically, there is no reason tubes should have higher distortion, and in fact the actual mathematical distortion series for a single-ended tube is less than that of a bipolar transistor! It is all about design choices and meeting listeners' preferences. SS designs tend to be differential, cancelling even-order distortion and leaving odd terms as dominate, so 2,4,6... HD is suppressed and 3,5,7 dominate. SS amps also tend to have higher feedback, which provides much lower distortion, but also means the higher-order distortion is more "present" as the feedback rolls off. Single-ended designs (tube or SS) tend to be dominated by second and third order distortion, gradually decreasing, while differential SS designs with high feedback tend to have very low low-order harmonics and higher harmonics, whilst still much lower than a single-ended design, may actually be comparable to the lower-order terms. But all usually inaudible, for either design, and again speakers usually dominate the distortion.
I have had and loved both tubes and SS over the years, and still feel some of the best sound from my old Maggies was using an ARC D-79 on top and Counterpoint SA-220 on the bass. But I also know the absolute performance was better using various SS amps. My current speakers (Revel Salon2) are a more demanding load and I am driving them with SS amps. There is also the reliability factor; my ARC I had to repair about once a year, and replace tubes every 3-5 years, while I have SS amps I have not touched in 10-20 years. And there's the heat from the tubes; nice in winter, vexing in summer. Modern tube amps have improved but I have insufficient experience. I will say that my old ARC SP3a1 tube preamp (modified) was very quiet and had as low or lower distortion than most SS preamps of the day (1980's~1990's).
5. Reliability, brand recognition and reputation, and so forth are important too, as are aesthetics. A basic black box is fine in my media room, but a pretty (to me) McIntosh component in the living room would be gorgeous. All else equal, or at least meeting my main requirements for power, low noise, low enough output impedance for the speakers, and reasonably low distortion, then looks, features, and warranty come into play. I put this 5th in line but it is not really 5th in importance, if only because so many good amplifiers can meet the first four criteria. Often enough there are a number of amplifiers that meet the technical requirements so I can decide based upon looks, price, reputation (including warranty), and features. Local availability used to be one of my main criteria, but that is harder now with fewer dealers carrying stock and in my area just fewer dealers period. I have driven 2-3 hours to visit a dealer having a product I want to hear, but more than that is usually not practical. I am fortunate to have a couple of dealers locally with decent stock of what I can afford, but the more esoteric brands I only read about on WBF.
This is already way too long and barely scratches the surface. I did not mention many of the technical details, nor did I discuss listening tests that I feel are very important to do in
my system (and much harder to do these days than in the past when loaners were routine; it is almost impossible to borrow an amp for a weekend from the local stores). But hopefully it's a start...
HTH - Don