When you get a credit card, the issuer agrees to let you use its money to charge purchases. At the same time, you promise to pay back what you borrow. If you don't keep your promise, the lender has the right to try to get the money from you. This can include eventually taking the money in your checking account. When you have a checking account at the same bank that issues your credit card, it might be even easier for the bank to tap your checking account for what you owe.
TL;DR (Too Long; Didn't Read)
If you default on your credit card, you are at risk for having the money in your checking account seized to settle the debt. Typically, the creditor will need to get permission to do this from the courts first.
When you put money in a bank, that institution has the right to take the money you put in your account to pay off any other obligation you have to the bank without having to go to court. This right, called the right of setoff, comes from the Federal Reserve and is how a bank can, for instance, wipe out your savings account to pay off a bounced check. As written, the regulation doesn't let banks use the right of setoff to take your checking account to pay a credit card debt. Because of this, some banks write their credit card agreements to get around the limitation and give them the right to take your checking account. If your credit card agreement has this language -- referred to as a security interest -- your account could be taken to pay off a defaulted credit card after all.
If you have your credit card and your checking account at different banks, your account is generally protected. Your credit card agreement with First Bank of Somewhere is private, so it can't just call Second Bank of Somewhere and ask for your money. Furthermore, your checking account agreement with Second Bank of Somewhere shouldn't allow anyone other than someone you authorize to take money out of the account. A court order can pierce this protection, though.
Collections, Judgments and Levies
If your credit card stays in default for too long, your checking account could still be at risk. Once the card holder gives up on getting you to pay the balance, it could choose to sue you for what you owe. If it sues you and wins, it can get the court's help to collect the judgment. One of the many tools the court can bring to bear is the power to levy bank accounts. With a levy, the court orders your bank to give your money to the creditor.
Your Credit and Checking Accounts
If you have a defaulted credit card, it could also make it harder for you to open checking accounts in the future. When you go to open a checking account, many banks and credit unions will check your credit. Having dings on your credit doesn't automatically disqualify you from opening new accounts, but serious problems could. Bankrate notes that banks are more likely to look at credit when you try to open an account with a linked debit card.