When you get turned down for a credit card because of bad credit, it's really not the end of the world. You still have many options, as you're about to find out.
Option #1: Your Credit Union/Bank
Most credit unions and banks now offer special credit cards for their customers. If you have a checking or savings account with a financial institution, check with them and see if they offer a credit card. If they do, apply for it. Chances are, you'll get accepted, albeit with a low credit limit, simply because you've proven yourself worthy based on your history with them. If, however, you've overdrawn your checking account, you may want to wait a few months and make sure you have a good history with them before applying.
Option #2: Local Stores
Generally speaking, getting a credit card from a place like JC Penney or Sears or Target is frowned upon because of the high interest rates these stores have for their credit cards, but for those with bad credit, they really don't have anything to lose by applying for a card with a store. In fact, it can be the start of re-establishing credit and getting a better card in the future. More than likely, if you apply for such a credit card, not only will you be accepted, but you'll probably get a decent credit limit. As long as you pay the balance in full, it'll be a good deal for you.
Option #3: Co-Signer
A lot of people with bad credit overlook one fact: if they get someone with decent/good credit to co-sign with them for the credit card, they'll most likely be accepted for even those credit cards that require good credit. Most people don't like asking someone to co-sign with them, as it can make them feel as if they're unworthy, but it is really a solid solution provided you're willing to make payments on-time every month on the credit card. If you have a spouse, getting a co-signer should be very easy.
Option #4: Secured Credit Card
Because of the fact that so many have bad credit, a lot of credit card companies now offer something called a secure credit card. A secured credit card is a credit card that is backed by the borrower's own savings account, so that if they spend too much, it comes out of their own money and not the company's money. While secured credit cards aren't for everyone, they're a great way of helping someone with bad credit to re-establish themselves.