Re-amping is a recording technique that allows you to dial in every possible sound after you have recorded your tracks. The trick of this process is to record the dry signal coming from your guitar or bass. This source signal can be routed back from your DAW to every possible amp and mic combination you like. So you are able to set the right sound for your mix after you have recorded a tight track. It is also possible to send the source tracks to a professional recording studio and get the sound of your favorite setup.
Usually you can divide the signal route into two sections, that can be successively recorded. In the input chain you record the source signal from your Guitar. It is important that you route the signal coming from your guitar or bass through a DI-Box. But it is also possible to use an instrument input of your recording device. The input chain looks something like that:
Guitar/Bass -> DI -> DAW
Theoretical this small setup is enough for recording you guitar or bass tracks and send them to a re-amping studio to get the desired sound. The problem with this setup is, that it is no fun to record your guitar without any amp. The simplest way to overcome this problem is to use an amp modeller in your DAW (like Revalver, Amplitube or GuitarRig) to get a reasonably sound during the recording process. The other way is to route the source signal to one of the free outputs of your recording interface and through a re-amp box (e.g. this one or a reversed passive DI-Box; that converts the low-impedance signal to a high impedance signal) back in your guitar amp and record it with a mic. This output chain looks like this:
DAW -> Re-amp Box -> Amp -> Mic -> DAW
Usually this routing is also used in a professional re-amping studio to get the right sound from your source signals. Another thing that is really cool on this setup is, that you can dial in the right mic position an experiment with different amps and setups without playing. So you are able to concentrate on the sound!