This is the single most difficult and risky thing to do. Mistakes are common to life. Everyone makes mistake; your boss no less. There are however some mistakes that can be detrimental to the growth of the company. In such cases where your boss does something or makes a decision that could put the company in trouble, you’re obligated to speak up because beyond the company being in trouble, your job could also be compromised.
The very thought of approaching your boss about something he didn’t get right is rather daunting. It could even be more terrifying depending on the type of person your boss is. Situations like this are less tense if the boss is open to positive criticism and is approachable. Nice boss or not, it just can’t be easy pointing out your employer’s faults. However, if it’s affecting the company something needs to be done so you need to figure out when and how.
Definitely not when the mistakes are minor things like grammatical or typographical mistakes (especially when they’re talking to or writing you or a few colleagues). Pointing out such mistakes could be embarrassing for your boss and might put your relationship with him/her in trouble. If it involves a larger crowd however you might want to get his attention to it.
Subtly suggest his emails or speeches be edited before they’re circulated. If possible suggest employing someone simply for the job (if no one is doing it already). Don’t point this out publicly either.
A miscalculation or a mix-up of numbers could be bad for the company’s finances. If your boss is making a mistake in this direction, it’s in your best interest to call him back.
Make sure you have the correct figures and can prove beyond doubt that he’s made a mistake. It’s best to approach him privately and on time too before the mistake gets out of hand.
Your boss might be displaying some attitudes or traits that you deem wrong. It could be his lackadaisical attitude to work, crossing the line with you, or something behavioral. Please be sure that this problem is in some way affecting your work before you go further. Also realize that there might be no-going-back after the next step.
In cases like this you might have to report to human resources or your boss’ boss (if there’s any). This is a sticky situation to be in and has every tendency to backfire. This is why you need to tread with caution. Make sure you have sufficient evidence to prove your claims. Remember if it backfires, you’ll have to face the consequences.
If it isn’t affect your work life and is not a threatening situation that is worth your career, I’d suggest you give it a rest. If you report your boss, regardless of if he is cautioned or not, you’ll always be known for bad-mouthing your boss regardless of the circumstances. Either way, be sure to think things through before making a decision. If need be, ask for someone’s opinion.