The most common vocal removal technique is based on the fact that in most songs the vocal is centred in the stereo spectrum, whereas a significant part of the other song components are not. It's possible, in fact a relatively easy operation in terms of audio signal processing, to compare the two channels of a stereo audio signal and remove what is common between them.
How well this technique works depends on the specifics of each song. Sometimes it does not work at all, either because much of the audio content is equal in both channels, so what remains is insufficient, or because the lead vocals are not precisely centred and so cannot be removed this way. This can happen for example when stereo effects (e.g. reverb or delay) were used or there was layering of multiple vocal takes that are spread over the stereo spectrum.
When that happens, filtering is the easiest next best thing, i.e. applying low and high pass filters excluding the frequency spectrum occupied by the song lead vocals. The problem with this technique is that probably much of the musical content of the piece will go along too. With experimentation it may be possible to narrow the filters in order to remove most (perhaps not all) of the vocals and still leave a useful, albeit not perfect, karaoke track. A filtering band from 200 Hz to 3000 KHz is generally a good starting point.
These two techniques can be combined to optimise results, i.e. only the stereo centred audio is removed within a frequency range.
You can try the combined technique with Audacity, an open source audio application, using the included effect "Vocal Remover".
Additional professional/industrial techniques combine these approaches with more sophisticated ones, like analysing in detail the frequency spectrum and targeting very specific frequencies, or removing spectrum components that do not remain very stable for a certain period of time (as the human voice has natural oscillations that most instruments don't have). These can be done by error and trial and targeting different techniques at different segments of the song to optimize results.