There are many different types of string, made of different materials for different purposes.
Main differencies are: material, thickness and tension, surface finishing, proportions of core and windings, production technology.
There are many related factors that are no less important for string sound - used scale, nut angle, saddle angle, nut and saddle material, frets material and profile, ...

Thickness sorting:

Acoustic, electric and jazz strings thickness:
Common string thickness - defined in inches: .008, .009, .010, .011, .012, .013, .014, .015.
.008 - .012 - elektric guitars
.009 - .013 - acoustic guitars
.011 - .015 - archtop guitars
Acceptable word labelling: Light, Medium, ... (see nylon string description).
There are many others hybrid sets of strings except listed thicness. Hybrid sets are usually composed of heavier bass strings and lighter treble strings.

Nylon:
The most of string producers don´t use numerical value for information about thickness. Commonly used word labelling::
Light, Medium, Medium Hard, Hard, Extra Hard
More accurate information about a string set is contained in tension description declared in kg or lbs.

Bass:
Common string thickness - defined in inches: .030, .040, .045, .050. Acceptable word labelling: Light, Medium, ... (see nylon string description).

Wound type:
Round wound, half round, flat wound, rounded with special material (nylon, bronze and nickel combination, ...).

Materials:
Gut - historic instruments, classic guitars
Nylon - classic guitars - black, white or gold nylon in combination with silver, bronze or brass winding
Bronze - acoustic and acoustic jazz guitars
Phosphor bronze - acoustic and acoustic jazz guitars
Nickel - electric, jazz and bass guitars
Steel - electric, jazz and bass guitars
Chrome - electric, jazz and bass guitars


Specials:
Nickel-bronze or nickel-phosphor bronze - double wound. Designed for using acoustic strings with magnetic pickup.
Electric, acoustic, jazz guitars
Nickel-nylon, chrome-nylon, steel-nylon - nylon wrapped regulal wound strings with a view of thickness rise without dramatic tension changes.
Electric, jazz guitars
Silk and Steel - silk core combined mostly with silver wound. Very soft and very loud strings.
Finger style, folk, ...