There are a great many schemes for sequencing DNA, but essentially all of the modern ones rely in one way or another on a variation of the dideoxy sequencing method. The concept was developed by Sanger and is wonderfully elegant and simple. In its simplest (but not best) formulation, one starts with a short piece of DNA that is labeled on the 3' end. This is hybridized to the piece of DNA that you want to sequence at a known position. Using a polymerase, the short piece of DNA is extended, copying the template DNA. However, in the mixture of nucleotide triphosphates that are being incorporated in the strand, there are a few that lack the 3' OH group on the ribose and therefore cannot connect to the phosphate of the next NTP. These dideoxy nucleotides effectively halt the chain extension at that point. Now, if you have in one tube a small concentration of dideoxy ATP and in the next ddGTP and in the next ddCTP and the last ddTTP, you can see that in each tube you will get a population of fragements of DNA. In the ddATP tube, all the fragments will end in A. In the ddGTP tube all fragments end in G, etc. By running these fragments next to each other on a gel, one can map the positions of each of the nucleotides by the size fragments that ended with that nucleotide.
In the automated sequencers common now, the difference is that the four different ddNTPs are labeled with different fluorophores. Thus, as the different sized fragments come down the gel, one just determines the color of the fluorophore attached to it and that tells you which base is next in order.