A burner is an inexpensive mobile phone that is designed for temporary use, after which it may be discarded. Burners are purchased with prepaid minutes and without a contract.
In actuality, although prepaid phones are often called burners, it's more likely that the handsets will be kept and the minutes topped up when they run out. Only people with extreme needs for privacy are likely to use actual burners. Many people became familiar with the concept of burner phones from watching The Wire, a popular television series in which drug dealers used burners to communicate and then got rid of them to prevent tracking by the police. Paying for the phones with cash rather than a credit card and having no contract with a service provider meant that there was no record connecting the user to the phone number to begin with. If the user suspected that the number was compromised they could just dump the phone and purchase a new one, which would have a new number.
These days, most relatively law-abiding people have smartphones, and cellphone numbers are increasingly being used as unique identifiers, connecting user data across multiple databases. Moreover, phone numbers aren't subject to privacy requirements in the same way that social security numbers and credit card numbers are, so they tend to be shared fairly readily. As a result, mobile number privacy is a growing security issue.
One simple method of increasing mobile security is the use of a burner phone app or service (rather than an actual device), which provides a temporary phone number that can be provided instead of the user's actual cell number.