All classes in the World of Warcraft are versatile, but each one has different roles in which it excels. It helps to choose a class based on the role you think you will enjoy playing.
Classes can be roughly categorized by whether they are front-line or support, whether they are best at melee (close) or ranged combat, and how easily they can be played solo (without needing to be part of a group for enjoyable play). The lines are not always clear-cut; for example, support classes can still be defined as primarily melee or ranged for the damage output they do, even though damage is not necessarily their strong point. There is significant overlap in abilities between different classes. Still, each does have its strong points, as outlined below.
Although people make alts in other classes, you may find that there is one particular class which you are called to, and that fits more strongly than the others with your particular psychology and playstyle. In this sense, Blizzard's advertising slogan for WoW, "What's your game?" is actually extremely appropriate. It may initially be a little difficult to figure out what your particular game is, in terms of class, but when you find it, you'll know.
New players should also remember that a given class's potency may not be obvious until later levels (particularly 40+). Sometimes all you need is a single spell that changes your gameplay from maddeningly difficult to excessively fun. Playing, for example, a Priest or a Druid might seem particularly difficult during early levels, but the class will suddenly change after learning a new ability or talent, increasing the fun and excitement of playing it by several notches. Having a source of advice on playing your desired class, from the beginning until the level cap (several go through various play-style changes, particularly depending on the talents available), can also make a huge difference in the degree of enjoyment you find in playing one class versus another.
Finally, players should understand that World of Warcraft is a constantly-evolving game. Any given class or talent tree's potency relative to any other may vary dramatically from patch to patch. The class descriptions below noticeably omit references to class balance for this very reason.