1. To put forth one's maximum effort. I've been beating my brains out trying to get a passing grade this semester, so failing by two measly points is incredibly frustrating. Don't beat your brains out trying to please these people—they'll never appreciate it.

2. To physically attack one, as with punches and other blows, such that they suffer significant injury. This phrase can be used both literally and hyperbolically. I'm worried that the captain of the football team will beat my brains out if he finds out that I'm secretly seeing his girlfriend. As if I didn't feel bad enough after losing the case, my boss beat my brains out for it once I got back to the office.