I can understand where you are coming from here - I am learning this as well, and sometimes it is really helpful to have a more relaxed explanation of something when it is new. Please keep in mind that I am only just learning this as well, but this is how I understand big-Oh:
It is part of asymptotic notation, which is used to describe the running time of an algorithm (i.e. it puts the running time into a range from best-case (Omega) to worst-case (big-Oh) and we can be sure that the algorithm will take at least Omega, and at most Big-Oh to run)
Big-Oh is the upper-bound, i.e. the slowest the algorithm will ever (successfully) run
O(n) is linear time - You can think of it as walking up a set of stairs, left-right-left-right. So you take one step on each stair (or loop through a list once)
O(n^2) is quadratic time - obviously this is much higher (will take longer) than O(n). You could think of this as taking n steps on each stair as you climb (or if there are 8 stairs, you would take 8 steps on each stair)
O(log(n)) is logarithmic time - this is the holy grail for many algorithms (like sorting; being able to sort in O(log(n)) is really good!). Trying to describe this in the stair analogy gets a little tricky, and I don't think I quite understand it well enough to try :) But, the general idea is that as the stairs get longer and longer, you don't have to do much much more work to climb them, because you are actually jumping up them
As to what is the base of the logarithm, it doesn't really matter for asymptotic notation, because with asymptotic notation you are trying to describe what will happen when the size (n) gets really really big (though it is often base 2). All log graphs (base >= 2) start to flatten out at high values, so they are always going to win at the top end.
There is a course on coursera, Design and Analysis of Algorithms, that has just started this week - one of the first lectures gives a high-level description of asymptotic notation, and the last question of week one's problem set made me think about running times and what beats what. It wouldn't take long to cast your eye over these.
Hope this helps a little!