John Wayne once said, "Never apologize. It's a sign of weakness." (As Captain Nathan Brittles, She Wore a Yellow Ribbon.)
But sometimes, an apology is a sign of strength.
One of the firsties (seniors) in my squad kind of screwed up. I approached him about it, and he fixed it immediately. But then I realized he had screwed up more than either of us had thought, so I told him about that, taking with me the handbook which showed how it was supposed to be done. He listened to me and saw the handbook, but ignored both. As a firstie, he thought he knew better than a two degree (junior) - after all, that's how it was always done. He said he'd "talk to somebody," which meant he'd tried to find someone with more authority than me to back him up. But the handbook has the authority, and everyone above me backs it up. I was ticked. I had shown him the right way to do it, but he simply refused to listen.
So he talked to that someone and found out that he was wrong. And then he came back and apologized to me, said he realized he had become the firstie he'd never wanted to be - the one who pulls rank and refuses to listen to two degrees. He thanked me for confronting him and apologized for being a jerk about it.
That's not an easy conversation to have. Personally, it's always tough for me to admit that I'm wrong, especially to a subordinate. It's a mark of his character that he was willing to admit his error and apologize for his attitude.
So I learned something about leadership today: apologies aren't always a sign of weakness.