The spine has three jobs
Structure: it holds your body upright against gravity. Movement: it lets you bend, twist, reach, and look around. Communication: it carries the nervous-system signal that runs every other system in your body.
When the spine doesn't move the way it should, all three jobs suffer. Pain is just the most obvious symptom, the deeper problem is that the rest of the system starts compensating.
Vertebrae, discs, and nerves
Your spine is built from 24 movable vertebrae stacked on top of the sacrum and tailbone. Between each vertebra is a disc, part shock absorber, part spacer, full of water and the cushioning material that protects the bone above and below.
Between each pair of vertebrae, a pair of nerve roots exits the spinal cord. Those roots connect to muscles, organs, skin, and everything else. When a vertebra isn't moving correctly, the nerve root nearby can be irritated, and the downstream effect can show up almost anywhere.
What chiropractors actually do
A chiropractic adjustment is a specific, controlled input designed to restore proper motion to a vertebra that isn't moving the way it should. It isn't 'cracking your back', it's targeted, and it's based on careful examination of which segment needs help.
Different techniques accomplish this in different ways: hands-on Gonstead and Diversified, instrument-assisted Activator and Arthrostim, computerized decompression for disc cases. The right technique depends on you.