Please explain 'log' to me

i'm trying to REALLY get into programming - beyond just knowing the syntax. i'm reading up on algorithms and data stuctures, but i keep running into some mathematical things i'm unfamiliar with. i haven't completed any algebra courses beyond intro to college algebra (somehow though, i managed to test out of intermediate algerbra and go straight to college algebra, anyways....). i see 'log' in just about everything i read.

i've been searching the internet, but since 'log' appears so often, it's hard to find out what it really is.

another thing i keep seeing is O and the text didn't really explain what O represented (unless i skimmed past it)

so, could someone please break this down for me (if at all possible without it's context)

O(n log m)