There are many types of reverb used in audio production. Reverb is an essential tool used by audio engineers to create space or spaces within a mix. They can define an entire sound of an album and create emotion within a song.
Digital Reverbs (Algorithmic and Convolution)
Digital Reverbs are created using mathematical equations to simulate realistic spaces like rooms and halls or unrealistic spaces like gated and reverse reverbs. Digital reverbs allow for more sonic characteristic control such as the size, decay and reflections of the reverb. You can also set the pre-delay which is the time space between hearing the reverb effect from the original sound.

Algorithmic Reverb
Reverb where the reflections are created by mathematical equations called “Algorithms” are called Algorithmic Reverbs. They can mimic real spaces and unnatural spaces. A gated reverb for an 80s snare sound is an example of an unnatural reverb sound created using an equation.

Convolution Reverb
Reverb where the reflections are created not mathematically, but by using samples (IR's, Impulse Responses) of an actual space. This creates a more realistic sounding reverb. There are several ways to create IR's such as capturing (using a high quality mic) a clap or a high pitched beep, or by sweeping through frequencies in the space. Special software is then used to extract the reverberation from the sound (Deconvolution), and the resulting IR can then be loaded into a convolution reverb plugin to be used in your projects. Below are two examples, "Open Air2" by Presonus and the popular “Spaces 2” by East West. Often, Algorithmic and Convolution reverbs are used in combination in music production.


Hardware Related Reverbs
Plate Reverb
This reverb was one of the first created reverbs in the 1950's. Plate reverb is created by sending a sound source through a metal plate (using a transducer) and capturing the reverberation it causes with contact microphones. The result is a spacious and bright sounding reverb. Today, these reverbs are created using emulations (Algorithmic, Convolution) instead of using the actual hardware as they were often quite large. Plates sound great on vocals but can be used for many applications in your projects. Below is an emulation of a classic steel plate reverb by UAD. You can view the EMT-140 hardware here https://en.wikipedia.org/wiki/Reverb_effect.

Spring Reverb
Spring reverb is very similar to plate reverb with the exception of using springs as opposed to metal plates to send sound through. Spring reverbs are well known on guitars and often found on guitar amps and were a cheap and space saving way to create reverb before the age of computers. If you've ever bumped or knocked into a classic Fender amp, you heard the result of the springs clanking around in the metal casing of the unit, sounding like a loud metallic thunderstorm…. scaring the crap out of you. Below is an image of a typical spring reverb tank.

(By Ashley Pomeroy - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=99049086)
Recorded Reverbs
These are reverbs that were captured by room mics such as those setup on drum recordings, to capture the spaces real ambience. These are then blended with the tracks or added to artificial reverbs and effects to get the desired sound. One way to create natural reverb effects is to play a sound source through a speaker in a space and capture the ambience with a mic or set of mics and use that to blend in with the original signal.