Stripe provides software that makes it easy for developers (and thus your e-commerce website or app) to accept credit card payments quickly. They define themselves as an: "instant-setup payment platform that would scale to any size".
You may consider Stripe's main competitor to be Braintree, owned by PayPal.
Stripe makes life easier for online retailers mainly because it accelerates the process of being authorized to accept credit cards online, which used to be a hassle for whoever wanted to sell online. Stripe turned this previously time-consuming process into just typing a little information into a Web page and waiting a few minutes. A no-brainer.
Thereafter, Stripe has methods in place to assure transaction security for both the consumer and the seller, and we've heard of perfectly legitimate merchants that have first been accepted and then rejected after a few days of using Stripe (and after launching their store).
Still, we think that services like Stripe and Braintree are currently the best methods to accept credit cards online for an e-commerce store or app.
At the moment of writing, Stripe's costs are 2.9% + 0.30 cents per successful payment with no refund or chargeback fees. This is currently the same pricing as PayPal.