I have a very good relationship with my manager but he is clumsy and takes details for granted. Today he sent me a spreadsheet and - by accident - he sent the budgeting allocations for 2014 in it (I was not supposed to see it as I'm a consultant/contractor and not an employee). The huge mistake he has done in the budget WILL affect the whole team including me. How do I go about telling him his mistake?
I would sit down with him in person (alone) and present the matter in the form of a question along the lines of " Hey "X", I got a copy of the budget emailed to me. Did you want me to do something with it?" That way you are being a "good" contractor by confirming that you are checking to see if he really DID want you to see it.
If he asks if you looked at, admit that you have and then point out the error. If he doesn't, then simply don't point it out. Obviously, his work will be checked by others, so the error(s) will be caught when it is. As a contractor, and not an employee, you have no involvement in the budget; nor should you want any.
How do I go about telling him his mistake?
If your relationship is good enough, it's simple.
You simply say "Hey, boss. You made two whopping mistakes today. First, you sent me your budget for 2014. But more important, [include the details that you think are problematic here]. Just wanted to let you know."
If you aren't sure your relationship is good enough, then just ignore it. Respond only if your boss realizes his mistake and asks you about the details in the spreadsheet. Technically, you probably shouldn't have read the details once you realized the spreadsheet wasn't supposed to be delivered to you.
Budgets are usually closely checked along the way, so mistakes like these will likely be caught by someone else. He might be embarrassed by those mistakes, but they aren't likely fatal.
In the corporate world, especially as you indicated with a political environment, "No good deed goes unpunished." That's especially true for good deeds that were unasked for. Relationships are more important than "helping when not asked to help." And relationships are based more on soft skills and regular "feel good" interactions and talking instead of "helping someone with a business matter".
You did work on something without being asked to do it, which - based on context - itself could be going out on a limb.
First let your manager know you got the email, and ask if he wanted you to look at it. If it could be a problem with him even sending it to you, then ask in person, very casually and quickly and privately.
If he says yes then go back to your desk and give it a reasonable but quick amount of time before you come back to him with your "discovery". Send it back in email with a caveat "I'm not sure if this is an issue but I noticed...is that what was intended?"
Focus more on dependability and productivity in regards to what your manager asks you to do instead of "being the teacher's pet helper." I don't know if that's where you are coming from or not, but sometimes the desire to "help when not asked" can be based on a lack of social/political awareness and wanting to "score points" with your boss, which in my experience can indicate a narrow view of relationship building.
Simple, save yourself and manager, it was sent to you so take the assumption that he intended to send and check the numbers and give him the feedback asking if he wants you to revise. Do it over email so that it is documented in order to avoid any accusations.