That's a big, open ended question. In very broad strokes:
Frame geometry affects fit, as well as handling, stability, and to smaller extent comfort over rough surfaces.
Fixies are bikes with a single gear hub that doesn't freewheel. There is no coasting because the turning wheel will turn the chain and turn the pedals. They are used for track racing, as training for some cyclists, as modes of transport for commuters, and as fashion statements for hipsters
Hybrids are a mix are a kid of cross between a road and mountain bike. They have a very upright riding position and are best suited for commuting, riding on pavement, crushed gravel bike paths, and possibly very smooth trails off road.
Road bikes are built for, you guessed it, riding on the road. They typically offer a more aerodynamic, bent over position than other bikes and are pretty much designed for speed.
Mountain bikes are a big category that encompasses several disciplines and bike types for said disciplines: trials, downhill, cross-country, technical, challenging terrain, as well as others
Cyclocross bikes are designed for the sport of cyclocross, a European concocted form of insanity and fun that takes place in the fall and winter and involves racing on grass, mud, snow, sand, and other "surfaces", as well as running up hills whilst carrying the bike, and bunny hopping over obstacles.. It involves getting really dirty and wearing women's clothes or teletubby costumes. Cow bells are required equipment for cyclocross fans, as are beer and frites. Cyclocross bikes can also be used for cross-country like trails and gravel roads. They are quite often used for commuting, and a number of riders use them because of their positioning which is a bit more comfy than road bike positioning,
I have no idea what conversions you're talking about, unless you're talking about van conversions. I think those died with the '80's. I guess that's a start. I'm sure others will add to this.