Guitar string thickness is measured by gauge
The gauge of a string is the exact measurement of guitar string thickness.
The high E string, for example, can be .010, .011, or .012 inches wide. You wouldn't think this tiny of a change makes a difference, but believe me, it does.
How are gauges of strings measured?
Guitar string thickness can either be written as a decimal of an inch (like .010) or more popularly as a number (like 10). For example, the numbers below represent the true diameter, in inches, of a set of 10's for electric guitar:
.010 - .013 - .017 - .026 - .036 - .046
Most guitarists simply refer them as:
10 - 13 - 17 - 26 - 36 - 46
Which numbers correspond to which string?
In the most popular electric guitar string set, the 10's as guitarists would refer to them, the gauges...
10 - 13 - 17 - 26 - 36 - 46
...correspond to these strings (this starts with the high E string, or the string closest to the floor, first):
E - B - G - D - A - E
Guitarists refer to packs of strings by the gauge of the high E string
Often, guitarists measure overall guitar string thickness of a pack of strings by stating a specific gauge of high E string. Bear with me.
Guitarists often say something like, "Can I get a pack of 11's for electric guitar?" The gauge of the high E string often governs the overall gauge of the other strings in the rest of the pack. Generally, if you pick a heavier gauge of high E string, than you're going to get heavier B, G, D, A, and E strings.
Let's get into the most common gauges of strings for electric, acoustic, and nylon string guitars.
Common gauges of strings for all types of guitars
10 gauge strings for electric guitar
10's are the most common string gauge for electric guitar. A pack of10's would have the following gauges:
10 - 13 - 17 - 26 - 36 - 46
11 gauge strings for electric guitar
This set of string thickness is more common for players who really beat the crap out of their guitars and want the strings to last:
11 - 14 - 18 - 28 - 38 - 49
11 gauge strings for acoustic guitar
11's are considered a light gauge for acoustic guitar strings, though I'm sure you can find lighter ones if you searched hard enough. A pack of 11's for acoustic guitar would have the following gauges:
11 - 15 - 22 - 32 - 42 - 52
12 gauge strings for acoustic guitar
12's are standard for acoustic guitars. A pack of 12's might often have the following gauges:
12 - 16 - 24 - 32 - 42 - 53
13 gauge strings for acoustic guitar
Some players prefer a heavier string on their acoustic guitars. They also may want to tune their guitars down to a lower pitch (a job well suited to heavier strings). A pack of 13's for acoustic guitar would often have the following gauges:
13 - 17 - 25 - 32 - 42 - 53
Classical guitar strings
Nylon strings are typically thicker than steel strings. Also, since classical guitar players are prone to take every single advantage they can get, I think it's fair to say that you'll see more variation in guitar string thickness with classical guitar strings in general.
The larger diameter of the string makes nylon strings a lot easier on the fingers when you're first starting to learn guitar. That, and it does sound more mellow when compared to steel strings. Here are the gauges of the strings for a nylon string set that is (fairly) standard:
28 - 32 - 40 - 29 - 35 - 43
Heavier gauges = harder strings to play
If you're a beginner, chances are that you might have the easiest time with a smaller gauge of string if you choose steel string guitars. The other option is to opt for the nylon stringed guitar. Though nylon stringed guitars have wider necks in general, the material of the string is more considerate of the fingers if you're just starting to learn.
If you decide to go with steel strings guitars, partially because that's the type of guitar you gravitate to, make sure to find one with a low string action. It really hurts to play any gauge of guitar string with high string action, although with time and guitar callouses just about anyone could get used to it.