What are people using for payment processing?

I suggest that you do some searching - this has been covered countless
times.

Sounds like you need to do some other base research though if you're not
aware of how payment processing works or even the basic requirements. There
is a pretty stringent security requirement depending on how you implement
it. Make sure you understand the implications of your design on the security
side of things or you'll eventually get hacked and then everyone will be
very, very sad.

On your last question, I wouldn't EVER consider allowing a user to submit
credit card data on a non SSL connection.

Your communication with the payment gateway does need to be encrypted also
but that's a gateway specific thing.

Good luck.