dubs got it right.. the first number is width of the tire in MM, the second is called the "series" and is the distance from the outside of the rim to the outside of the tire as a percentage of the width. the third number is the rim size. if you were trying to figure out a tire height by hand...
tire size 265/60/18
265/25.4 = 10.4 (tire width in inches)
10.4 x 0.6 = 6.2 (size of ONE sidewall)
10.4 x 0.6 = 6.2 (size of other sidewall
18 (size of rim)
add up the heights
6.2+6.2+18 = 30.4 inch tire
here is how everything affects your handling
width (first number) a wider tire will improve cornering ability, more rubber touching the road gives you more traction, but a wider tire is more prone to tracking groves in the road and losing traction in icy or snowy conditions
height (second number) the larger your sidewall, the more body lean you will feel during cornering, a taller side wall will also give a softer feeling ride. sidewalls that are 30-50 are usually considered low profile, 55-70 are pretty standard.. and anything over 75 is usually a pretty tall sidewall
tread pattern, a directional or asymmetrical tread pattern will give you better handling, directional (also called rotational) will give you the best traction, but are also a pain because they have to turn a certain way, so when you go to rotate the tires, they must be rotated front to back, a rotational spare tire will only work on 1/2 of the vehicle (examples, goodyear eagle GT, michelin hydroedge)
asymmetrical just has to be mounted a certain way on the rim, once its properly on the rim, it can go anywhere on the vehicle (example, falken zeix912, goodyear eagle response edge, nitto invo)
speed rating, speed rating is a rating that tells you the tire can go XX speed for 10 minutes continuously without failing.
S -112
T - 118
H - 130
V - 149
Z(or W) - 168
ZR - 186
for a normal person.. the speed isnt what matters, the speed rating will also tell you how well constructed (stiff) the tire is. the higher you go the stiffer the tire, a higher speed rating will give you a stiffer ride, less body lean in turns and generally crisper steering
the generally accepted standard is you can deviate from the factory rating by 1 rating, so in the case of the VX which came factory with H rated tires, anything T or above would work.