Many biological questions can be translated into a DNA sequencing problem. For instance, if you want to know the expression level of a gene you can: copy its mRNAs into complementary DNA molecules, sequence each of the resulting DNA molecules, map those sequences back to the reference genome, and then use the count of alignments overlapping the gene as a proxy of its expression (see RNA-seq). Other examples include: determining the 3D structure of the genome, locating histone marks, and mapping RNA-DNA interactions. A not up-to-date list of biological questions addressed by clever DNA-sequencing methods can be found here.
Typically, the wet-lab scientists (the people wearing white coats and goggles) will design and perform the experiments to get the sequenced DNA samples. Then, a bioinformatician (the people using computers and drinking coffee) will take these sequences --encoded as FASTQ files-- and will map them to a reference genome, saving the results as BAM files.
Going back to our gene expression example, this is how a bioinformatician would generate a BAM file from a FASTQ file (using a Linux system):
STAR --genomeDir path/to/reference/genome --outSAMtype BAM --readFilesIn my_reads.fastq
Where STAR is a spliced-tolerant aligner (necessary for the exon-intron junctions that may be present on the mRNA).
PS: Once the mapping results are obtained, the creative part begins. Here is where bioinformaticians devised statistical test to check whether the data is showing biologically meaningful patterns or spurious signals born out of noise.