I read that Mac OS X and bsd are related. How closely are they related. Can Mac OS X software be tweaked and installed on BSD?
The Wikipedia BSD article is good (and accords with my own understanding, for what that's worth). It says that Darwin, the system on which Apple's Mac OS X is built, is a derivative of 4.4BSD-Lite2 and FreeBSD, and notes that 4.4BSD is the last release that Berkeley was involved with.
So, Darwin is as BSD as you can get (just like all the other BSDs!). OS X refers to those parts of the distribution which aren't open-source, principally the GUI, but including a variety of frameworks, and anything which relies on these won't be portable.
OS X as a whole is a UNIX 03 system. That's equivalent to being a truly POSIX-compliant system (as opposed to being POSIX-like).
As other answers have noted, the userland parts of the OS are unsurprising to anyone with much unix experience, and I've rarely had any difficulty building portable-unix software on OS X.
In contrast, the non-userland parts of the OS are pretty different. Apple seems to be willing to innovate in those areas fairly cheerfully. I think (but I'm not positive) that these changes are formally part of Darwin. One of the most obvious differences is that launchd has replaced cron, at, inetd, and much of the startup infrastructure.