Their complexity depends on the model.
There's a Russian model on eBay right now that simply tells you if you're ok or if you're in trouble. Some will actually measure the amount of radiation.
In its simplest form, you turn it on and point it at the object of study. If it makes a click (or a beep) every once in a while, you're safe. If it clicks A LOT and CONSTANTLY, you've got a problem.
I used a Geiger counter as part of my laboratory checks when I was assigned the task of "radiation safety monitor." And it was frequently used to check the outside of packaging containing radioactive material to see if anything had leaked in transit.
Ah, those were the days. Toxic chemicals and radioactivity all around me.