Payment Gateways are third party organisations which allow you to accept credit and debit cards on your website.
There are a number of options available, some of which I have outlined below. If you have any questions or queries, let me know.
PayPal
Paypal is one of the most popular mayment gateways available, there are a number of different integratino options but generally all will allow you to accept Visa, Visa Electron, Mastercard, Maestro, Solo and American Express. Implementation generally doesn't take more than one day to set-up, so for those in a hurry it is at the top of the list. PayPal are probably the world's most widely known payment brand, although they have come under considerable critism by users.
I recommend PayPal for people starting out with eCommerce, it's a cheap way to set-up your store and very fast to set-up without much hassle.
SagePay
Sage Pay Go is the complete card payment service that takes a bit more time to set-up and requires more details aabout you and your organisation. Sage Pay Go with Form integration is a method of passing transaction details from your website to Sage Pay, in order to obtain authorisation on those details. Sage Pay Go with Direct integration is designed to enable you to take card details on your own secure servers and pass them across to us for authorisation and secure storage in a server-to-server session that does not involve redirecting the shopper to the Sage Pay payment pages.
I recomment SagePay for those taking a larger amount in transactions with an established business online.