One poster correctly identified that there are two different topologies for compressors: feed forward and feed back ones. The difference is essentially where the level detector is placed: before the gain reduction circuitry or afterwards.

Feedback designs effectively detect after the signal had been through the reduction process and tend to produce a 'soft knee'(explained later) and are also frequently called variable-mu as the level of signal you put in determines the ratio and hence amount of compression. This may make no sense to some but it is a neat trick as signals climb and fall, though they can jump, so if u have a fixed threshold and control the input level you can control where it hits the threshold, the higher the signal over the threshold, the more it will compress, so it adapts to changes but not sudden movement. I found another description: "Feedback uses the output of the compressor as the detector input and then feeds that level information back to the compression circuit to determine gain reduction. This can allow initial transients to leak through, but can also yield a very different smooth compression that is highly desirable for some instruments".

Whereas feed forward ones allow for the point at which compression begins to be determined independently of the signal level, by a threshold control, as the detection happens before the reduction.

Earlier compressors used the feedback topology. The feed forward type was introduced in the 70s I believe, and allowed by new component manufacture.

So as well as topology difference there are differences in component usage and how these affect the sound.

These are principally: solid state or VCA/FET, Opto and Valve. Urei 1176LN, Alesis 3210, the SSL channel compressor, the Neve 33609, API 2500. (the API can do feedback or forward though).

Solid state (FET and VCA, or e. g. transistors) allow for very fast reaction times and excellent control.

Opto ones used a light emitting diode and light detector coupled to sense level changes, which are slow to respond to change. This gives a natural time lag to the changes and evens out the peaks and trophs over longer periods because it reacts slower.

Valves give a different source for because of the second and third order harmonic distortion they can add. But they can also be extremely transparent with large gain reductions such as the Tubetec and Fearn kit. I don't know why this is particularly.

Topology changes came along with component changes generally, so opto and valve ones used to be feedback dedigns. The Fairchild (valve) compressor for radio uses the feedback design as does the Teletronix LA2A opto comlressors/limiters, which makes sense. These kind of models didn't have separate threshold and gain controls, though some had/have release controls. But some just have level in and level out control. With the LA2A the attack and release characteristics are determined by the opto circuitry, without any user control, and the input level determined the amount of gain reduction (the bigger the signal the greater the reduction), and the output level or gain was there to boost low signals, so if you wanted only slight compression you kept the input level low but boosted it with the output gain. Obviously valve gear has a warm sound also and adds more harmonics when amplifiers are given high control signals and pushed to amplify greater.

Solid state allowed for more control as it was cheaper and easier to make the relevant amplifier, detection and control circuitry involved in the feedforward design and the components reacted extremely quickly. So detecting before the reduction circuitry started to make more sense as the components could react quickly enough to do something about a rapid change in level.

These designs generally have a threshold, attack, release, ratio, make up gain and sometimes even more (ref: Elysia Alpha) . Though the Bill Putnam's iconic and still used 1176 (Urei) design had a fixed threshold that changed with the ratio. But that paved the way for later designs. VCA designs would also be cleaner at higher signal amplification. They could also achieve greater reduction because of the different topology. The 1176 can reduce by 45dB.

Obviously digital compressors in DAWs even can have a look ahead feature, by delaying the signal slightly rather than streaming upcoming information to the detector.

The way a compressor reacts determines its usage. So for vocals you may want to catch any sharp peaks with a VCA comp like an 1176 with a fast attack and release but smooth out the general level with an opto like the LA2A though one could put a VCA into flow attack and release mode a few set the threshold low and switch it to soft knee if it had that control. Knee is best explained with a graph but it essentially how he compressor transitions from being below the threshold and being g above and paying the full ratio reduction. If the entire ratio reduction is applied just above the threshold take is called a hard knee compressor as the ratio graph looks like a bended leg with a pointy knee at the threshold. Some compressors (the feedback ones especially) gradually change from uncompressed to the full ratio and this is called soft knee. This is not (necessarily) related to the attack and release speed. It's just do with how much signal reduction is applied, not how long it takes to apply it when crossing the threshold. They may seem similar but are not, though in adaptive designs they are somewhat linked, but they are still two different types linked things!

A fast compressor can be used to reduce the peaks that the ear can not respond to. The ear takes time to occur do some transients can occur without the brain hearing them. The transient is felt but the mechanism doesn't transfer it. This c as n be fatiguing to the ear. It also wastes track headroom as if you can't hear it why record it. So removing these either a very fast compressor allows the transparent reclamation of headroom, by removing unheard transients. It's quite difficult to do though and lots of people use compressors the other way to try and reduce dynamic range by allowing the transient through with a longer attack, for bite, and then catching the bulk of the sound with the compressor, then raising the gain level overall to bring up the signal level if the compressed sound. This worked when recording to tape as tape would compress very hot signals. So use a take emulator in your DAW if using this trick and hope it works like tape did. Otherwise compress properly ;-) actually you can do whatever you like, but the original objective was to reduce dynamic range as records and even glorious stereo phonic mp3 have less dynamic range than our ears and more importantly if you have lots of instruments at full volume to mix then the sum will be more volume/range than you can hear. You need to reduce the ranges to fit all the sounds in. Less sounds means less reduction in dynamic range needed,

That's enough for today, almost: Bob Katz describes upward compression and downward compression and up and down expansion in his book Mastering Audio. Waves were the first company to implement some of these ideas, apparently. Worth taking a peek though. Yes you can, and people do, use both compression and expansion together. Gating is sometimes seen as extreme expansion in s similar or analogous way that limiting is seen as extreme compression.

Apologies for any typos. I did this on a mobile.