There are a couple of ways you can do this.
1. If you have a merchant account with your bank - contact your bank they can usually get you started. They will provide you with a couple of options to allow you to accept credit card payments and your web developer can hook it up for you. It means that any payments you get go directly into your bank account and you usually have access to a portal that will allow you to see your transactions.
2. Third Party merchants - these are companies like PayPal. They will accept the payments on your behalf, then you need to transfer your money to your bank account.
Both options have charges and you'd need to work out what works best for you.
If you go through your bank - they usually require you to meet certain requirements, eg. have a SSL Certificate, specific terms & conditions, policies on returns etc
Some Customer may prefer to pay you via a well known Third Party merchant and some may feel wary of purchasing products if they don't know your site.
If you have any questions regarding credit card payments, please let us know on email@example.com.