Power stays on after turning car off

Just this evening I turned off my car (2000 Toyota Corolla), went inside and then came back out an hour or so later to discover that my radio, clock, and cigarette lighter still had power. Without the key in the ignition, I'm still able to listen to the radio, for example. This was never the case before. I had the battery replaced five days ago, but I doubt that makes a difference....Anybody got any ideas how I can fix this so that my battery doesn't drain.
I'm completely car illiterate, so talk to me as if I'm dumb. Because I am.
I'm completely car illiterate, so talk to me as if I'm dumb. Because I am.

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