Bash colon operator in variable substitution?

This takes the variable branch_name, if it is defined. If it is not defined, use HEAD instead.

See Shell Parameter Expansion for details:

3.5.3 Shell Parameter Expansion

The ‘$’ character introduces parameter expansion, command substitution, or arithmetic expansion. …
The basic form of parameter expansion is ${parameter}.
…
When not performing substring expansion, using the form described below (e.g., ‘:-’), Bash tests for a parameter that is unset or null. Omitting the colon results in a test only for a parameter that is unset. Put another way, if the colon is included, the operator tests for both parameter’s existence and that its value is not null; if the colon is omitted, the operator tests only for existence.

${parameter:-word}

If parameter is unset or null, the expansion of word is substituted. Otherwise, the value of parameter is substituted.

Substrings are covered a few lines below. The difference between the two is

${parameter:-word}

vs

${parameter:offset}
${parameter:offset:length}

${parameter:offset}
${parameter:offset:length}

This is referred to as Substring Expansion. It expands to up to length characters of the value of parameter starting at the character specified by offset.
…
If offset evaluates to a number less than zero, the value is used as an offset in characters from the end of the value of parameter. … Note that a negative offset must be separated from the colon by at least one space to avoid being confused with the ‘:-’ expansion.