How do the proteins derived from DNA translate into certain traits? For example: How do the cells know, when producing a liver, when exactly to stop growing in a certain direction?

I've been told over and over again "well it's like a computer, so the little 1s and 0s blah blah blah" or some vague shit like that. Please explain to me, in no uncertain terms, how exactly a pattern of Thymine, Adenine, Guanine, and Cytosine translates to gills or something. Don't dumb it down so much that I won't get an accurate picture, I'm genuinely curious and I'm tired of the bullshit my bio 100 teacher keeps spewing.