CoreCommerce offers 4 PayPal payment gateway options:
PayPal Website Payments Standard is the basic type of PayPal. When this is enabled, the customer will have to fill out your store checkout and will be able to choose PayPal Website Payments Standard in the Payment section. When they are sent to PayPal, they will be able to log in to their PayPal account or use a credit/debit card without having an account. The line items will be displayed on PayPal and the order will be submitted on their end, not your store's.
PayPal Express Checkout requires the customer to have a PayPal account. When you enable this, it will display on both the View Cart page (as a button) and on the Checkout page (as a choice in the Payment section). When the customer is sent to PayPal, the line items will not be shown. The customer will be sent back to the store to choose their shipping method and will submit the order from the store's checkout page.
PayPal Website Payments Pro is considered a payment gateway. You would enable this is in the Credit Cards section of your Payment Settings. You would use this to accept credit cards on your site, and the customer would not know that PayPal is being used. When you set this up, you can either use the API signature or certificate. If you are going to also use PayPal Express Checkout, you will need to use the certificate.
PayPal Payments Advanced is the newest type of PayPal integration. With PayPal Payments Advanced, you can accept both PayPal and credit cards on your site. You can set it so that customers are sent to PayPal to pay, or you can give them the ability to pay with a credit card without ever having to leave your site. This type of PayPal will work with all other PayPal types, and since it's not a payment gateway itself, you can have it enabled alongside a true payment gateway like Authorize.net.
In the CoreCommerce Admin Panel, you will find these options under Settings > Payment Settings