A “burner phone” is a cheap, prepaid mobile phone that you can destroy or discard when you no longer need it. In popular media, criminals often use burner phones to evade detection by authorities. You might use a burner phone for privacy reasons, as a last resort, or during an emergency.
A “burner SIM” is a related term, and refers to a cheap, prepaid SIM card that you can insert into another phone. You may plan on only using the SIM card for a limited period of time and not linking it to your real identity.