General:
* Error
* Blunder
* Mishap
* Fault
* Slip-up
* Oversight
* Miscalculation
* Misjudgment
* Misunderstanding
* Flaw
* Defect
More formal:
* Inaccuracy
* Misapprehension
* Aberration
* Deviation
* Discrepancy
* Misconception
* Fallacy
Informal:
* Boo-boo
* Goof-up
* Screw-up
* Flub
* Faux pas
For specific types of mistakes:
* Logical mistake: fallacy, misconception
* Technical mistake: error, glitch, bug
* Moral mistake: transgression, sin, offense
* Social mistake: faux pas, blunder
The best synonym will depend on the specific situation and the desired level of formality.