Discuss one type of DNA sequencing and the steps involved in that method.

Probably the easiest method to understand is the Sanger method for DNA sequencing. In this case some standard techniques regarding DNA polymerase chain reaction (PCR) are used. The main difference here is that you must use a single stranded template to ensure you have the correct sequence.
First, we must understand what the key components are:
DNA polymerase - an enzyme that works under specific conditions that binds nucleotide bases to a growing DNA strand on the 3' end based on a template it is running along.
DNA Primer - a short strand of DNA that is hybridized to the target strand (already complementary to the target strand) that serves as a starting point for sequencing. You need this otherwise your DNA polymerase won't be able to construct a new DNA strand.
Deoxynucleotides (dNTPs) - Standard deoxyribonucleic acid bases: deoxyadenosine, deoxyguanosine, deoxycytidine, deoxythymidine. These are the common elements of any DNA strand.
Di-deoxynucleotides (ddNTPs) - These are the experimental add-ons. These are the standard 4 bases (A, C, T, G) but without a hydroxylated 3' end. This is important because once one of these bases is added to the DNA strand, synthesis is terminated.
Now, you put these elements together in a buffer with a single strand of DNA that you want to sequence. The primer binds to one end of the DNA template. The polymerase then can bind to the template and primer and start constructing a new DNA strand. The polymerase randomly selects normal deoxynucleotides and dideoxynucleotides and produces DNA until there is no primer left, there are no dNTPs left, or there are no ddNTPs left. Of course, in order to allow a full strand to be sequenced, there is a significantly lower concentration of ddNTP in the buffers than normal dNTPs. Also, you have to have 4 different buffers running at the same time (one with each ddNTP) or you must otherwise have a label for each ddNTP for an automated sequencer.
Classically, you use 4 different solutions, one for each dideoxynucleotide base, and each of these bases has some sort of label, such as radioactive phosphate or a fluorophore. You can then put a portion of each mixture into an electrophoresis gel (which sorts strands by size), and perform electrophoresis. The gel is then exposed to a photographic film, which is then developed to show where each A, C, T, and G are. Then, it is a simple matter of writing the letters from top to bottom!