Stereo and channel groups on a mixer are types of buses. In an audio sense, a bus is simply a collection of signals. Each channel strip can be assigned to one or more buses on a mixer so equipped -- some small mixers may have only a master stereo bus. There is no difference between the master stereo bus and a pair of group buses. In both cases these are simply collection points for audio intended for output from the mixer.
In live music, the stereo mix is the logic destination for all audio coming in to the mixer. Instruments and microphones are processed on each channel strip and the output of each is sent to the stereo mix. The output of this mix then goes to external amplifiers and finally to the house speakers to provide the performance sound the audience enjoys. Every mixer, large or small, is capable of operating in this manner, whether there are channel group buses or not.
Most medium-to-large mixers have some sort of group bus arrangement, most commonly four or eight buses. These can be used alone or in stereo pairs. Bus assignment is made for each channel strip, usually with buttons above or along side the channel faders. It is common to combine multiple audio signals from a group of instruments, for example a drum set or a horn section, by using a group bus. Channel faders balance the component instruments, are sent to a channel group bus, which in turn is routed to the stereo mix. In this way, the level of all the drums or horns can be controlled with just the channel group fader.
When using a mixer to supply a multi-track recording device, it is common to use the channel groups to supply the recording device, while using the stereo mix as a monitor of the overall recording. Using a four-bus mixer to record a string quartet might see each instrument with its own microphone, and each mic routed to a separate channel group, then sent to discrete tracks on the recorder. Group buses are made for flexibility. There are no right or wrong ways to use them.