The application will accept payments from users with paypal's Payments Standard. All of this money will be stored temporarily in the company's paypal account, and then redirected to some of the users involved in the transaction via their paypal accounts (we'd also keep a cut of this money). Would this implementation require a banking license or any other kind of license?
EDIT: region = California
I'd appreciate any help.
It's very much region-dependent. Better ask a lawyer.
Banking license certainly not since you're not opening a bank. Look at many freelancer sites which act as intermediaries forwarding customer money minus fee to contractors. The same situation like you need.
Anyway, there may be some regulations which are country or region specific. Maybe you need to pay certain taxes dependent on your location or location of your servers. Or maybe you're not allowed to have servers with personal customer data outside of your jurisdiction. Or something else.
Ask a lawyer. It's a complex matter.
Try asking here: http://startups.com/
Sounds like an escrow situation. It looks like California has some information here: http://www.corp.ca.gov/FSD/escrow/default.asp