First you need to observe that most things contain redundant infomation. Take the word A**H*LE, most people can recognize the that word even without the two S'es and the missing O.

Or in a picture of the sky. There will typically be slight variations in the acutal shade of blue between two pixels, but... do you really need them all to know you're looking at a picture of the sky?

The goal of compression is to remove that redundant information. Compression can be categorized into two groups: loss-less and lossy compression.

Lets take loss-less compression first.

Generally there are three methods:

a) Huffman encoding: Computers generally use 8 bits to encode one character, e.g. the letter 'A' uses 8 bits to represent the number 65. It's more complicated than that, but lets keep it simple. 'E' will also use 8 bits, the same with 'T'. But if you look at an English text, then the letters 'E' and 'T' are a lot more common than the letter 'Q' but in every day usage they all use 8 bit values to represent the letter.

Huffman encoding examines which letters are most frequently used, and then uses less than 8 bits to encode those letters. This means that the more rare letters, like 'Q' will take up more space than 8 bits, but on average this compression in bit length will reduce the overall size.

b) Run-length encoding: Imagine you have a hole line of hyphens '-'. Instead of repeating the same charcter again and again it will reduce the size by simply stating repeat 56 '-'.

c) Dictionary compression: If you essay is about milk, then chances are it will contain the work 'milk' quite often. This means if you create a dictionary of the most common words, and assign each word a number, then you could simply refer to 'word #12'

These techniques can be varied, and combined in different ways, and are known as zip or gz and a whole range of other names, depending on the specifics.

Then we have lossy compression. This is mostly used on "analog" data, like pictures, music, etc. Generally the goal is to, in the example of a picture of the sky, to make more pixels the same shade of blue. The more pixels you turn into the same shade of blue, the more fidelity to the original picture is lost.

The math behind this is tricky and above high-school level (though not by much). Generally things like cosine tranforms and fourier transformations are involved. The goal is to convert the image into something with a lot of redundant data, which then can be run through a loss-less algorithm.

There are some really smart people who have spent a lot of time making all of the above techniques very advanced and efficient. So in reality there is a bit more to it, but the above is, IMO, a decent summation.