32", 29", 28/27", 700c, 650b, 26" are the common "adult" sized bike wheels.
* 700C and up are all 622mm diameter rims.
* 650b is 584mm, and is one type of 26" wheel. There's more wheel and less tire, making it a good hybrid, cross, or road tire for small to medium sized people.
* 26" is generally 559mm mountain bike tire. It's a wider tire measured in decimal inches (vs fractions). This is common for small adults or big kids.
* Anything smaller is a specialty kid size
The inside width of the rims range from 13mm to 53mm. Rims can take tires from about 1.1x to 2.5x without special considerations.
Some tires are not the size marked. I have some 32mm tires that are actually 27mm wide.
Generally, we use "clincher" tires, which have a lip on the inside edge of the rim that catches a wire "bead" embedded in the inner rim of the tire.
These can be normal, or tubeless. Tubeless have special rim tape to make the rim air-tight, and a sealant is loaded into the tire after it's installed.
There are also "tubular" which have the inner tube sewn into the tire, and the tire is glued onto the rim.
These have a higher inflation pressure, which can reduce rolling resistance (to a point).
They're more costly, require special rims, and generally only used in racing, where every gram of weight counts (no wire bead).
Road - the top tube is parallel to the ground. The head tube and seat tubes are closer to vertical (around 72 degrees). This makes them better for going up hill (forward CG). Handlebars are drop-bars, meaning they are flat across the top, then at the widest part, they curve down and back. The frame has no suspension. The cranks are attached to 2 or 3 chainrings, with the largest being 48 teeth or larger. The rear cassette is usually 7-11 gears, with the largest being 28 teeth or smaller. The smallest rear gear is usually 11-13 teeth. Some older bikes might have 5 or 6 gears on the back. 8-10 use the same hub, and are more versatile. Wheels are generally 650b or 700c, with inside width around 13mm. Often, they have deeper rims to give an aerodynamic advantage to the leading edge. Tires are generally 21-27mm. Common weight is around 16.5 pounds.
CycloCross - A road bike designed for a little more rough terrain. The top tube may slant down a couple of degrees toward the seat tube. Fork "rake" and frame width will be slightly wider to allow up to a 37mm wide tire. Often, these have enough tread to make it through light sand, rocks, or mud. Common weight is around 20 pounds.
Touring - A road or hybrid bike loaded up with bags and comfort accessories so that it is convenient and comfortable for very long distance rides.
Hybrid - generally a mountain frame without suspension. May also be called "Urban", "Comfort" or "Cruiser". Technically can be used to classify any non-standard combination of mountain and road parts.
Mountain - 26", 650b, or 29" tires are common. Handle Bars are typically straight bars, but may curve up and/or back. Top Tube angles down to about the top of the rear tire. Seat and head tubes angle back more, such as 70 degrees, moving the CG back. Also, CG is lower. This helps in descending rough terrain. Tire widths are commonly 48mm and up. Often, these have at least front suspension, and often rear suspension as well. Subcategories include "All Mountain" (heavier parts for more durability) and "Downhill" (more slack tubes, maybe as far back as 68 degrees). Front chainrings are generally single or double. Commonly a 32-36 tooth front ring is used. Rear typically ranges from 14-34 teeth.
SS - Single Speed - single speed bike, most commonly a hybrid or mountain bike. Single on the front is generally 32-36 teeth. Rear is typically 14-18 teeth. Tires are usually 40-55mm wide.
Fixie - a single speed mountain or BMX. Commonly has rear brakes activated by pedaling backwards. Sometimes used interchangeably with Single Speed.
BMX - rigid frame, with the seat tube stopping very low. The top tube and seatstays generally follow the same line. The seat post may be tall-ish, but usually, the rider won't be on the seat. Fat tires, no suspension, designed for aggressive dirt tracks, jumps, etc. Tends to be smaller. Handle Bars are usually risers, meaning they curve up substantially. The frame is usually somewhat short.
Chromoly / Steel - Primarily iron alloys. Tube ends are usually cut to mate, and a lug (matching socket) is used to hold the pipes together. Brass solder is used to braze them together. Often ornate, due to being hand made. Tube sizes are generally thinner, but moderately heavy weight bike. Common for older bikes, or custom built bikes.
Aluminum - generally 5000, 6000, or 7000 series aluminum. Lighter weight, stiffer frame, but thicker tubes. Possibly more prone to cracking under very heavy abuse than with a steel bike. A good choice for inexpensive bikes.
Carbon Fiber - Carbon Fiber Reinforced Plastic. Made of 2-part epoxy and carbon fiber in the same manner as "fiberglass" composites. Generally more expensive. Absorbs vibration. Can flex without fatigue cracks. Exceeding ultimate loads cause catastrophic failure rather than bending. Generally has a large variety in tube sizes, reinforcing around the bottom bracket to reduce crank flex, but thinner on the top tube to reduce weight. Uses friction paste rather than lubrication where seat tubes are installed. This is the de-facto gold standard. Expensive due to the amount of manual labor involved.
Titanium - Titanium alloy. Very light weight, but more risk of metal fatigue cracks. Generally better for lower-mass riders. Rides like a cross between aluminum and chromoly. Very expensive due to limited availability of the metal.
Road - designed for smaller gap between gears. Less concern with mud clearance. More concern with weight.
Mountain - designed for wider range of gears. Designed for low speed, high torque. Designed for mud clearance (CX and MTB). More concern with reducing chain slap.