Scientists have been exploring DNA sequencing methods ever since DNA was first discovered to have its unique double helix structure of sequenced bases. Credited with the discovery of the double helix form of DNA are scientists James Watson and Francis Crick, from England's Cambridge University, in 1953. They used clues and evidence gathered from other scientists, and together with one of them (Rosalind Franklin) received the Nobel Prize in 1962 for the double helix discovery.
As science evolves, so do DNA sequencing methods. In the beginnings, with less advanced technology than we now have, DNA sequencing was more crude and laborious; for the first couple decades DNA sequencing was through a process called Chromatography, which, simplified, means separating mixtures of substances into more basic parts, whose colors can be seen as photographically different in color.
In the early to late 1970's, more rapid newer methods of DNA sequencing were invented in quick succession, and many new methods since then have been created. Beginning in the early 1970's, Frederick Sanger and colleagues of the University of Cambridge founded a series of methods called 'Chain Termination' methods. They were quickly replaced in popularity by the Maxam-Gilbert DNA sequencing method, which was invented by Allan Maxam and Walter Gilbert around '76 - '77, based on first chemically modifying DNA and then cleaving it at specific bases. This method became popular because more pure DNA could be used, rather than having to clone read-starts to produce a single-strand DNA as needed to be done using Sanger's chain termination method.
However, that quickly changed around as the chain termination methods were made more efficient, and using less toxic chemicals and lower radiation than the Maxam-Gilbert method. Other less popular methods were discovered during those times but were either too labor-intensive, inaccurate, time-intensive, or simply too complicated to catch on widely. Since then, many more DNA sequencing methods have been invented, most of them derivations of Sanger's chain termination methods. They include terms such as Dye-Terminator Sequencing and Shotgun sequencing (a method used for large genomes).
Below: Different methods create different looks; here are samples of a variety of DNA sequence image outputs for different processes.
Other newer methods are called High-Throughput sequencing, In Vitro Clonal Amplification, Parallelized sequencing, Sequencing by Ligation, and current technologies studying DNA sequencing using mass spectrometers in various ways. The current race is to build a device and method that can do DNA sequencing of entire genomes very quickly, accurately, and at an economical price. The first DNA sequence cost an estimated $3 billion, and as of this writing the current cost is around $50,000. Companies are racing towards technology and machinery that will reduce the per-sequence cost to under $1000, and when it reaches that affordable level DNA sequencing will become a routine medical test just as an MRI or CT scan