Stripe is a payment processor for online payments. They are a company which take those credit card details you enter and actually take your money on behalf of the shop.
Apple Pay is supported by Stripe, so it's a non-competitor (Apple Pay is just a way of giving your card details to a processor).
PayPal is the closest competitor, but their APIs are quite old (although improving) and when Stripe launched PayPal's pricing was very cmoplex. Also people just don't like the PayPal model for business.
Square is a payment processor, mostly, for physical payments mostly seemingly for hipster coffee shops. They do online payments, but nowhere near the same level Stripe do them.