Why do we use DNA sequencing methods such as shotgun?
This is a technique from the "old time" of genome sequencing. The underlying method for sequencing is the Sanger chain termination method which can have read lenghts between 100 and 1000 basepairs (depending on the instruments used). This means you have to break down longer DNA molecules, clone and subsequently sequence them. There are two methods possible.

The first is called chromosome (or primer) walking and starts with sequencing the first piece. The next piece of DNA (which is  and then uses a specific primer from the end of the sequence. The next (contigous) piece of the sequence is then sequenced using a primer which is complementary to the end of the first sequence read and so on. This technique doesn't require much assembling, but you need a lot of primers and it is relatively slow.

To overcome this problem the shotgun sequencing method was developed. Here the DNA is broken into different pieces (not all broken at the same place), cloned and sequenced with primers specific for the vector used for cloning. The leads to overlapping sequences which then have to be assembled into one sequence on the computer. This method allows the parallelization of the sequencing (you can prepare a lot of sequencing reactions at the same time and run them) which makes the process much faster and also avoids the need for sequence specific primers. The problems are to sequence repetitive sequences, as overlaps are not as clear here. To resolve this problems a first draft is made and then critical regions are resequenced using other techniques as primer walking. Have a look at the Wikipedia page about shotgun sequencing if you want to read more details.