I’m not sure anyone has offered a super complete answer, to let me expand on Paintcharge’s summary.
There are three primary types of clubs:
Woods
Irons
Putter
Woods are used for hitting the ball a very long way. A wood is distinguished from an iron by having a longer shaft, as well as the characteristically oversized striking head, both of which add to the distance the ball will be struck. The “number” of the wood describes the angle the face of the club is set at; the lower the number, the lower the launch angle of the ball, and the further it will go. (Despite the name, they aren’t made with wood anymore.) The very lowest angle wood is the 1-wood or more commonly called the “Driver” and has a club facing of 9 to 13 degrees; this is the club you use when you want to hit the ball as far as you can. A 2-wood has a slightly higher angle, a 3-wood slightly higher, and so on. If you buy a proper set of golf clubs you will usually have a driver (marked “1”) and a 3-wood and a 5-wood.
Irons are used for other types of shots that are not putts. They have shorter shafts than woods, and the characteristic metal, angled clubhead. As with woods, an iron’s number is proportional to its angle. A 3-iron is angled at 21-24 degrees, a 9-iron at about 45-48 degrees. A club angled even slightly more than a 9-iron is called a “pitching wedge” but you could call it a 10-iron because that’s basically what it is. Irons with even more extreme angles can be purchased and are usually used for very short shots or for shooting a ball out of sand (the “Sand wedge”) or some other unpleasant place. If you buy a proper set of clubs you will usually be equipped with irons numbered 3 through 9. 1 and 2 irons are for sale but usually not in a set, and an amateur doesn’t really need them; a wedge is a useful buy, though. Some sets come with a pitching wedge.
The purpose of different angles, as others have pointed out, is that it allows the golfer to take a full and proper swing while directing the ball a different distance. It is much easier for me to take a proper swing with a 9-iron to strike the ball 130 yards than it is for me to use a 3-iron and try to hit it with about 70% of my usual force.
Furthermore, the higher the angle, the less the ball will roll when it strikes the ground, obviously. As you get closer to the target, you are less interested in distance and more interesting in placing the ball on target. If you are very far from the hole, further than you can hit a ball, all you want is to shoot the ball as far as you can without veering off the fairway, so you hit it with your driver and hope it rolls a long way if it lands in the fairway. But if you are just 60 yards away, what you’d like to do, ideally, is drop the ball near the hole with very little roll; to that end, a very high angle of flight is ideal, since it minimizes roll. How a golf ball bounces and rolls after landing is a bit more random than how it flies, so it’s best avoided if you’re looking to be accurate.
As Paintcharge points out, “short distance” in golf is a really long way. The meek little 9-iron can easily hit a ball over 100 yards. My 70-year-old mother can use her driver to hit a ball close to 200 yards, which is further than any major league baseball player will hit a baseball this year. Golf balls are designed to really fly a long way, so the necessity of increasing the angle of the clubhead so you aren’t continually blasting the ball over the green onto the other side is rather central to the design of the equipment. There’s no rule saying you don’t play a whole round of golf just with your driver, but making short shots would be maddeningly frustrating.
Of course, the ability to select the angle of shot can sometimes serve other useful purposes. If for example you are at a distance that would usually call for the 9-iron, but it is obvious that striking the ball at that angle will cause it to hit a tree, perhaps in that case you do want to use a lower iron but try to hit it a bit more softly, thus avoiding the tree. Or you may be in a situation where a water hazard looms ominously between you and a distant hole; shooting with a long range club might present more of a danger that the ball might unexpectedly roll into the water, but shooting with a higher iron will cause it to safely come to rest in front of the water where you can take a second shot to get over it.
The putter, as has been pointed out, is meant to simply roll the ball on the green, and has a perfectly flat angle.
There are other types of clubs, too, called “hybrids,” that sort of blur the line between woods and irons.