A disc galaxy is a spiral galaxy.
There are four types of basic galaxies.
Elliptical (anywhere from a giant sphere to an american football (or perhaps the ball in rugby, if that's more familiar).
Spiral (rotating disc of stars with a central bulge)
Irregular (well, the name kind of says it).
and Dwarf (which are far, far smaller, about 1/100 of the Milky Way, and they are also classed by the above three cats)
We know the Milky way is a spiral because of observations and mass distribution. We've also managed to figure out things like our orbital velocity. Technically, we're a barred spiral galaxy. The latest news is that we're almost 50% bigger than we thought, based on a new orbital velocity of about 914,000km/h.
But yeah, figuring out what the home looks like when you can only see from inside one room is a touch difficult.
Check out these two wikis:
Milky Way - WikipediaGalaxy - Wikipedia