Recordings
Recording is simply converting sounds to a recording-medium, in this case, a computer hard-drive, and finally to CD or DVD. This can be done in a studio, a nice location, or live during a gig.
In a studio situation, the acoustic space is tightly controlled - the instruments and voices are recorded up-close, and any effects, such as reverberation, are added at the mixing stage.
In the studio, bands can either record 'all at same time' (this is called live recording), or build up a song layer-by-layer, using overdubs. The live method retains the spontaneity of live playing, and is quick -- but it has less control over individual instruments and voices than the overdubbing method. Normally, three, or four takes will be edited together, as needed, to create the finished track. With the overdubbing method, the band members play to a guide track, building up the track, part by part. This means mistakes can easily be fixed, by replacing any parts that have wrong notes, or bad timing. Of course, with all the run-throughs of the track for each band member, plus the fixes, means that the tracks take much longer to finish by this method - but the plus side is much greater control of the sound, and the ability to fix any problems. As a useful alternative, just vocals and solos can be overdubbed, which can combine the best of both worlds.
Location recording is the best method for choirs, orchestras, and brass bands. The sound of a large space with a good bunch of musicians is not really attainable in studios (unless there is a big hall or church attached!) So . . . the musicians set-up, and play. The music is recorded, again with a few takes, to be edited later. The Overdub method can also be used, if the band is small enough, and there is time.
Live Gigs are still location recording -- but it's also a concert, so there are no extra takes, or overdubs possible! However, the rehearsals can be recorded, and these can act as another take, if a small mistake needs to be edited.
An amplified gig will be recorded from the stage, while using a PA system. All the channels, for each instrument and singers, will split between the PA & Monitors and the recording rig, so that the PA mix won't affect the recording. In addition, audience mics will be used, to capture the ambience and rapturous applause . . .

