Reverb

Reverb is actually the same thing as delay, only based around much shorter time frames - so small that they are imperceivable. This emulates the real-world phenomenon where we hear sounds from not only their primary source but reflections off most nearby objects and structures. These reflections are very finely delayed compared to the sound that follows a direct, line-of-sight path from its source to our ears. The sound may reflect many, many times before actual reaching our ears, causing the reverb of a sound to last far longer than the source's original duration. For example, if you clap in a cave or empty parking garage or church, you can hear the reverb from the sound for several seconds although the clap itself only occupies a very brief moment in time.

Most reverbs are designed to emulate fairly reflective environments like halls, small or large rooms, and caves. Early attempts to recreate these used metal plates or springs inside amplifiers or external hardware. Some became quite popular, and many digital reverb engines now offer plate and spring algorithms, in addition to room, hall, and cave simulations.

Common Controls

Time/Decay - Controls how quickly the reverb decays, simulating the absorbancy of materials in a real-world environment.
Pre-Delay - The time between when the dry signal hits the reverb and when the reverb actually produces output. This simulates large rooms such as a concert hall, where the reverb has a much larger distance to travel than the line-of-sight path, creating a noticeable lag between the direct sound and the reflections.
Mix - Sets the relative levels of the dry signal vs. reverb. Most reverbs will diminish the dry signal as you go from 50% to 100% until only the reverb is audible.

Advanced Features

Tone and Dampening

Real-world materials do not reflect sound perfectly. Some materials absorb some of the sound, absorbing certain frequencies more than others. This contributes to a certain tone to the reverb. Reverb units may express this as a tone knob, or simply have Bass/Mids/Treble knobs. Basically, they are simply filtering the output of the reverb to darken, brighten, or mid-focus its tone to simulate the frequency-dependent absorbtion of real-world reverb.

Some reverbs offer dampening which darkens the reverb as it decays. This simulates the real-world phenomenon where lower frequencies travel long distances with less attenuation than high frequencies. But it can also help get a sound similar to a fairly absorbant room, where the high frequencies are absorbed faster than the low frequencies. I like to use dampening to get a more natural tone.

Some take dampening in a different direction, giving the reverb a blooming quality, where it takes some time before the higher frequencies resonate and hit a peak before decaying.

Modulation

Adds a chorus-like modulation to the reverb but not the dry signal, making the reverb sound more lush.

Particle/Octo/Pitch/Shimmer Verb

Various manufacturers have created methods to liven up reverb in complex ways. They typically seem to involve high-feedback, short-time delays, pitch-shifting, and modulation, perhaps with some filtering and compression to keep things from getting too crazy. It's difficult to completely understand everything that they do, and it's difficult to recreate them from individual effects, due to their complex feedback networks and their exact internal signal chains. Most of these are designed for very ambient uses, where the verb has space and time to "swell" over its lifespan.

Freeze

There are some reverbs that seem to freeze whatever sound you play infinitely. They typically do this by setting the time/feedback to 100% while you hold a footswitch.