A convolution reverb takes a sample from a real-world space, usually called an impulse response or IR for short and uses this to digitally simulate the reverberation of that space.
Convolution reverb takes an input signal (the sound to be reverberated) and processes it with the sound of an actual or virtual acoustic space to create the illusion that the input was recorded in that space.
The sound of the acoustic space is captured in an impulse response (IR), which starts as a recording of a short sound such as the firing of a starter pistol or the bursting of a balloon in the acoustic space in being recorded. This initial sound excites the reverberation (the response) in the space, and so the impulse response sounds like an explosion followed by the reverb reflections of the space.
The trigger sound is usually a somewhat broadband sound like a shot or balloon pop, because the impulse response recording should capture the widest range of frequencies in the reverb reflections possible for as accurate a recreation as possible. And, of course, for stereo convolution reverbs, you need to capture a stereo recording; for surround reverbs, you need a surround recording.
Once captured, depending on the convolution reverb software in question, the IR recording may need to be processed to remove the original impulse trigger sound and leave just the response, a process called deconvolution. Many convolution reverbs, however, don't require this step.
Once you have the IR for the space as a file, you then load it into the convolution reverb and input your sound to be processed. At this point, the software convolves the two digital audio signals together to create the output.
The advantages are they are great for replication of real-world spaces to get an almost true representation of the reverb characteristics of that space.
Infinite number of possible rooms
Naturalistic through very high density of reflections
Can create very realistic rooms
Most transparent and controlled reverb, perfect for adding subtle room space to your sound
Good for special FX and acoustic reproduction
Record and use your own IRs
Depends on good quality IR to create good results
No modulation over time as with good algorithmic reverbs, can sound static and cold
High CPU usage