What happen if the Bitcoin network is down, or all miners stop keeping records? I am not sure I understand Bitcoin very well. What I understand is that Bitcoin is there because of the network.

What happens if the network is down for any reason?

What happens if the miner that keeps my records of Bitcoin dies or doesn't want to open his computer again?
Nothing happens. The longest chain of blocks in existence, so long as it builds off the latest checkpoint) is always the valid chain, no matter what. So long as at least one node brings it back eventually, there's no problem. (And you can keep it yourself if you're particularly worried about it. If you're running the standard client, you already are.)

Unlike a bank account, your Bitcoin balance doesn't have to exist in any particular place to be valid. The miners create the mathematical relationships that prove it's valid such that anyone at any time can tell that it's valid just by looking at the mined data.

If you connected to any Bitcoin node right now and presented a longer chain of blocks (so long as it included the last checkpoint) than they already had, they would switch over to your chain of blocks and it would become the official chain. It wouldn't matter that prior to introducing it, you had the only copy. It's mathematically valid, so the network would accept it.