Second Chance checking accounts are designed for people who are not allowed to open normal checking accounts due to past banking behavior or bad credit. The reasons you may be disallowed vary, but once you build a bad history, you're reported agencies or networks like Chexsystem. When you request to open an account, banks and credit unions check these databases to see if you have a bad record.
Overdrafting a couple times doesn't get you on these lists -- usually a prolonged history of delinquency is required -- so don't assume that you have a disqualifying checking record. But if you have trouble opening a 'normal' or 'free' account, consider second-chance checking accounts.
Second-chance checking accounts are designed to help you develop a better checking record. They usually have a monthly fee of somewhere between $5 and $10 and offer a short list of features like check-writing, a debit card, and online banking. They don't include premium features like free checks, debit rewards programs, or overdraft protection.
Basically, it's never the best checking account that a bank or credit union offers, but if you open a second-chance account and keep a good record, the financial institution usually allows you to "upgrade" to a different checking account within 6-12 months.
When you have to open a second-chance checking account, your relationship with the financial institution isn't 'normal' -- you're not a 'standard' customer. So it's in your best interest to stay in strong communication with your financial institution, letting them know your intentions of being a good checking customer, and immediately contacting them when anything questionable occurs with your account. The goal is to develop a checking history that leads the institution to believe you're a reliable customer as soon as possible, so that you can enjoy the features most customers enjoy.