The effect of the output switching from left to right speakers is "Panning".
Some mixers have a PAN control on each channel to allow you to set that channels postitioning within the Stereo arena. I've seen this used to quite good effect where more than one DJ/presenter/speaker (human not speaker cabinet

) is present - the person with the mic on the left of the stage has their channels PAN set may 30% to the left, and the person on the right...their mic channel is panned 30% to the right... it gives the person speaking a bit more "prescence" and, whilst still clearly audible through both left and right speakers, is less loud through the speaker furthest from them - which helps, I think, with making the scene more natural.
Auto Panning is a feature which can be used on music tracks for effect from time to time - much in the way you've described above - I only use it at clubby type nights and birthday functions for younger (sub 21) crowds - the Auto-pan effect syncs to the BPM of the playing track and you can set it to bounce either the whole music, or just certain freqeuncies on-the-beat, from left to right to left to right etc... Best used in short bursts covering maybe 4 or 8 beats.
Panning/auto-panning features on many of the effects units which are available seperately from mixers, and can be a nice effect if used sparingly.