I understand that a transaction is a record of 1+ Bitcoins being given from one party to another.

I understand that a block is a collection of unverified (pending) transactions that gets verified every ~10 mins or so.

I understand that mining is the act of running specialized software that will solve some cryptographic problem, and as a result, allows a block of pending transactions to be added to the block chain (history of all transactions, ever).

So first off, if anything I have said so far is not true or misleading, please begin by correcting me!

Assuming I'm more or less correct, I'm still not seeing why mining is crucial to the system, and am (obviously) not seeing the forest through the trees here.

If I send 10 BTC to someone, why does someone else (a "miner") need to verify this? Shouldn't this transaction be stored in a DB somewhere, and verifiable by querying that DB?