An account at your financial institution (be it Bank, Credit Union or whathaveyou) where you store a supply of money to cover for (to back up) the Checks/Cheques you write.
To recap: A Check, in the monetary sense, is a promissory note from one party to the other. When you write a check you promise the receiving party that they can then redeem this useless scrap of paper (the check) for a different type of useless scrap of paper (the money). The money that answers for this writ is stored in you checking account, and when the payee cashes the check, money is deducted from you checking account to be given to them.
Checking accounts are also becoming useful now-a-days as you can have money directly electronically transfered in and out of them use EFT or similar system, in which case it's just like writting a check except no scribbling is involved and it happens nearly instantly. This is useful for many things such as, my favorite, paying bills.
When you run out money in your checking account and some tries to redeem a check you have written, it usually won't work due to Insufficient Funds, which is a polite way of saying you're out of money. In this case the check either bounces (it is made invalid, usually incurring a $20 returned check fee and an irate payee), or you Bank/Credit Union will extend you a line of temporary credit to cover it -- usually at a very high interest rate, but it's a lot better than a bounced check. | 
