Can I use my device on a different carrier?

Sometimes it's desirable to use a device on a carrier that does not officially sell it. Often this is due to regional differences and travel (for example, using a US phone in Europe) but also because some carriers simply do not offer phones that other carriers do. How can I determine if it's possible to use my phone on a different carrier, and which carriers it is compatible with? ----- *This community wiki is intended to be a canonical answer to a common question. If you feel that improvements could be made then please edit the post(s).*