A flag is also another name for a status bit, i.e. a binary toggle reflecting some aspect of the status of a system.
Take, for example, the carry flag. If an addition performed by a microprocessor or arithmetic logic unit causes a carry, then a particular bit will be set. If the addition does not cause a carry, that bit will be cleared. The value of the bit, then, reflects the condition of the result of the addition.
A typical ALU will have a zero flag, a carry flag (which normally also serves as an inverse borrow flag), sometimes a sign flag, sometimes a (signed) overflow flag, and others.