Term encompassing all the weird and wonderful higher-dimensional theories that theoretical physicists can come up with. Can refer to the concept of there
being further spatial or temporal dimensions beyond the normal three spatial ones - ie can be lots of dimensions of space, or dimensions of actual space plus
that of time. Not a term used by theorists, apparently - more of a popular term.
As well as the spaces which have been imagined by mathematicians for some
time, of geometry in however many dimensions, can refer to Kaluza-Klein theory, supergravity, or superstring theories. These latter theories address
deficiencies of the Standard Model, although that isn't necessarily wrong, with
the concept of supersymmetry. Adding squarks, sleptons, etc. to the mound
of itty-bitty particles floating about.
But many physicists like them because they are rather beautiful theories, as they
make things simpler by adding dimensions.
Cool Implications
With several spatial dimensions, you can imagine us corresponding to
two-dimensional people living on a surface in a three-dimensional world, as in the
book Flatland. If you think of the universe as analogous to a doughnut (it isn't though, as the universe is flat), but in four dimensions - a four dimensional doughnut with a
three dimensional surface perhaps - and small enough, you might see somebody
looking just like you, with their back turned to you. If you put your hand on their
shoulder, you would feel a hand on yours - the person ahead of you is yourself!
Anyway, I find stuff like that cool. I read a lot about that in Hyperspace by
Michio Kaku. I hope I understood it enough so that what I just said was vaugely
correct.