What would happen if I logged in, and then before browsing to a different page on the site, deleted the cookie?
You would no longer be logged in.
After I log in to a website, does it store some kind of cookie in my browser, and every time I navigate to a different page on that site, it checks the cookie for my identity?
Yes. Most likely, you are dealing with a "session-cookie". These cookies do not store any information themselves, but use a long string to identify yourself to a server. I would suggest doing some research on cookies. As for the (I'm guessing assumed) question of "Why cookies work on different pages?" is because cookies are tied to the domain, and not the exact URI.
Cookies contain names, values, and expirations (along with a few others). The most common you'll see are sessions, which use an identifier to load a session-state from the server containing your information. These are the safest cookies as everything is centralized and not as prone to hijacking. The other kind is a regular cookie, which has a limited size and stores information client-side. Anything that has to do with shopping or anything that tracks users most likely uses sessions, while something like a customizable javascript-y page probably uses a normal cookie. The former tracks information server-side for additional security, while the latter poses no security risk, and leaves the information for the client to manage.