Somebody who teaches guitar told me that 1234 means down down down down, and 1 and 2 and 3 and 4 and means down up down up. But why? What do these numbers represent?
If you are playing a song that has four beats in a bar (it's in 4/4 time), you can easily strum on each beat of the bar with a down-strum. By using only down-strums you can play steadily, with an even volume. If you want to play twice as fast (in other words, play eight strums per bar), you can play in between the beats with an up-strum. Your right-hand moves at just the same speed, but because you are now strumming up in between your down-strums, you are playing twice as many strums, and twice as fast.
In his book The Ultimate Guitar Tutor Tom Fleming describes this as "The Puppet Rule". If you tap your foot on each beat of the bar, you play as though your strumming hand is connected to your foot with a piece of string. So, when you tap your foot you play a down-strum, when you lift your foot you play an up-strum.
The music below shows how this works with an Em chord. In the first bar there are four down-strummed 1-beat chords. In the second bar there are eight chords, which are twice as fast and alternate up and down strums.
Some extra bits to add to Bob's answer.Unless you're pretty good at changing chords, don't bother trying the second bar strum pattern, as the last up-strum gives you half the change time that you had in the first bar. A lot of self-taught guitarists get round this by doing the up-strum with open strings.In some keys you can get away with it. In most, it sounds naff and amateurish.
When you've mastered both patterns, try to vary things by strumming, say, 1,2,3and4. Or 1and2 3 4. There are very many combinations - just play the ands, and you've got a reggae type rhythm. Don't forget that you could leave out one or more of the down-strums, etc, to give a total of absolutely loads of different strum patterns, without changing the up/down movement of that arm.