No, they're completely unrelated things. As you say, an unlocked device means it's not locked to a particular carrier: you can insert a different SIM card to change to a different carrier, and it will work. A locked device will not connect to another carrier if you change the SIM card.
The bootloader is the first thing that runs when you turn the phone on: it's responsible for loading the core Android code, so that Android can load itself and run. A locked bootloader uses digital signatures to verify the Android code it's loading, which means it will only boot a ROM (firmware image) that's signed by the device manufacturer. In contrast, an unlocked bootloader will load any ROM. Unlocking a locked bootloader will wipe all user data (but not the ROM) from the device. This is a security mechanism, so you can't get at the data on a phone by installing your own ROM on it.