First, nearly all of these cable types are some variety of "unshielded twisted pair" or UTP cable; this refers to two twisted pairs of comer conductors, wrapped up in a polyurethane cladding. UTP uses several properties of electromagnetic propagation to send signals, by varying the voltages on the cables differentially. There are other types of cables (fibre optic, and shielded cable are two), but unless you need their specific benefits, twisted pair copper is the most cost effective. (Fibre optic offers more bandwidth, but is significantly more expensive, and somewhat more fragile, shielded cabling adds expense, but not performance, and is mostly used in places where the electromagnetic transmissions from networking cables can cause problems.)

Of twisted pair copper cables, there are a couple of subtypes. Stranded is the most common type - if you plug a network or Ethernet cable from your router to a computer, this is the likeliest type. It's made up of strands of copper wound together in an insulator; over longer distances, it loses its signal carrying capacity, it has the advantage of being less expensive and flexible and finally it's easier to torque or bend stranded cable to meet a particular requirement.

Solid cable uses one larger wire in each insulating run; it has significant electrical performance issues, meaning it will in this context transmit data further with fewer losses. Most of the dedicated wiring in buildings that have network wiring built in is using solid cable. If you've ever had to pull stiff cable through walls or a drop ceiling, this is the cable type you've had hands on experience with.

If you've heard of CAT5 cable (or CAT3, or CAT5a) and had someone selling you CAT6 or CAT7, you're seeing one of the major differences between stranded and solid cable right there - transmission distances. Transmission distance for a cable goes up with the capability of the cable to carry electricity - thicker cabling carries electricity with fewer losses over a given distance; better insulation also improves transmission distances. While running multiple cables together (like stranded cable does) can approximate what solid cable does, it's not as effective or robust, though flexibility and reduced expense are good benefits.

CAT 6 cable is CAT5 cable with better shielding and a thicker wire; CAT 5a is CAT5 cable with the same shielding as CAT6. CAT7 cable is rated to a thousand meters (one kilometre!) and is thicker still and has higher end insulators. Note that neither CAT6 or CAT7 are actual recognized standards - this sort of cabling is, like 'draft-n' wireless, something where the hardware vendors are making a product in advance of the actual standard being finalized, though parts of it have been released.

Both CAT6 and CAT7 cables are considerably stiffer, and harder to run around corners, as you might expect and they are more expensive.

The appropriate question is "What sort of cabling do I actually need for this project?" You can save considerable expense by only using the actual cabling you need, rather than just running CAT7 cabling for a bunch of 10-meter lengths where it offers no benefit over cheaper runs.