I find my coworker's mistake, would I report to our boss?

It would depend on the seriousness of the error. Most things should just be corrected after you let them know of the goof, no need to get the boss involved. Running to the boss with every little thing will quickly make you a lot of enemies.

everyone makes mistakes. would the co-worker cover it for you? if so, I wouldn't rat them out. I did see that this will cause big losses to the company. if it can't be fixed before the losses occur maybe you won't have to tell the boss, let him see for himself that your co-worker messed up.

It's a hard thing to decide well I will not tell the boss right away talk to your co worker that he have to fix it before the boss will know. Sooner if he will not do anything about it the company/boss will conduct investigation if it's really a big loss of the company then he will be the one to fired your co worker not you.

Just find all the proof first, and if it's really serious, then you can talk to your co-worker, trying to advise him/her that it was a really big mistake to do. who knows, actually he/she didn't even know that it was a mistake.