Measuring is an absolutely fascinating procedure. The predominant method is called sequencing by synthesis. Illumina is the company that basically rules the market for this technology, they have some decent explanatory videos if you’re interested.
The first steps are to isolate DNA, then break it into small bits, replicate those small pieces thousands of times, and then anchor these to a glass slide. Then flow a solution over the slide with the individual DNA bases (ATCG) that have been modified to fluoresce when illuminated with a certain wavelength of light. These fluorescently labeled nucleotides get assembled one by one into strands that match the small fragments from the DNA sample. Each time the fluorescent DNA bases are added a laser is shined on the glass slide and a very sensitive microscope camera reads how many strands shone at the frequency that corresponds to A T C and G. This tells you which base just got added to which strand. Repeat a couple hundred times and you have a huge number of sequences of ATCG’s corresponding to each short strand of DNA in the sample.
The problem now is assembling these short sequences into a full genome. Basically, the puzzle is finding the arrangement of short sequences that maximizes the overlap between them and their agreement with a reference genome. It’s a pretty interesting problem that (I believe) is classically solved with dynamic programming, although I don’t know the most recent methods.
I really don’t think it’s well enough appreciated how integral computation is to DNA sequencing. Signal processing in the camera, assembling the genome fragments, determining whether a mutation exists, and all the subsequent analysis to give biological meaning to a sequence.
Edit: editing is a whole other, extremely complicated procedure. In fact, direct editing isn’t really possible now. Instead we can basically just delete and insert.