What is the concept of predestination? Are there different denominations that believe in predestination but have differences in what they believe about it? If so, what are the differences?
The concept of Predestination in so much as it speaks to the concept of Election is rooted in a couple of different passages in scripture. Romans 8:29-30 being the foremost. It basically says that he knew from the beginning who he was going to choose and he "Predestined" or choose them. There are many other passages that support this concept, outlined in this answer.
The question of whether he has predestined every event that will happen in the world is a slightly more contentious one. However, I believe quite strongly that he has indeed predestined every event that will happen. This does not mean that our actions are meaningless or that we should simply drift through life. Romans 8:28 (NIV) gets to the heart of this
As for denominations that believe in predestination to one degree or another. I know that Reformed traditions (Presbyterian, URC, CRC, Reformed Baptists and countless other "reformed churches") ascribe to some tenet of this belief.
I cannot speak to other traditions right off hand, but those who know of them are welcome to add them.
Single Predestination or Double Predestination?
Now before you start thinking "Aw, man! There are two of them?!?" Let me explain the difference:
God chooses us, solely by his grace, to go to Heaven. God does not choose people for Hell. If we go to Hell it is because of our own sinfullness.
God has chosen some people to go to Heaven, and some people to go to Hell.
Now that we know what they are, what does the Bible teach?
Ok, that seems to indicate that we are Predestined to Heaven, now what about Hell?
Based off these verses, it seems clear to me that the Bible teaches Single Predestination and not Double Predestination.
God has predestined everything. Here's a simple explanation:
A man was put before two doors, and given the free will to choose a door. He chose one and opened it. Behind the door, it said, "Chosen from the beginning of time".
See, God knows what we will do and yet we are given the free choice. It's hard for our finite minds to understand this concept, as we are limited to what we perceive here and now.