I have not flown on planes very much in my life and the ones I have taken have been non-stop trips. I am looking at prices for a trip I'm going to take alone and the cheaper ones are stopover flights which make the most sense to purchase. I want to fly from Chicago to Seattle and the stopover would be in San Francisco. This is probably a very moronic question, but when I check in at Chicago, do they give me 2 boarding passes? One for Chicago and one for San Francisco? So when I get off in San Francisco, I just walk down the terminal to my next fight and hand over the other boarding pass when I get on?

The short answer is yes. There are a few exceptions, but if you book a flight that includes a change of planes, when you initially check-in, the agent will tag your bag to your final destination and give you boarding passes for all the legs of your trip.

But, when you arrive in SFO, double check to make sure the gate for your connecting flight hasn't changed since your boarding pass was issued.

I am assuming that the two flights are on the same airline?

Same flights need not necessarily be on the same airline to get 2 boarding passes, but the 2 airlines do need to have a close working arrangement (i.e. code sharing)

For example, Chicago to SF may be on AA (both an AA flight # as well as an AA plane). The SF to SEA flight might be on an Alaska Airlines plane. Those flights happen to have both an Alaska as well as AA flight #, so you will be taking an AA "flight", you will get boarding pass issued in Chicago, but you'll be going to an Alaska airlines gate and flying on an Alaska plane.

This can seem confusing to infrequent fliers but is really a non-issue.

J62's absolutely right. But now, even if you're on two airlines that are not partners, you can still do online check-in on both separately and have boarding passes in your hand before you leave home.

In fact, back to the OP's post. Nowadays, you don't go check-in at the airport and handed boarding passes. You check-in online as early as 24 hours before your first flight, and print all your boarding passes.

If you have luggage to check, you can still drop it off at a counter or curbside. If you don't, you just go straight through security to your gate, without having to stop at a check-in counter.