This should help to get you started.

We use Fintech at work. It is a program by which someone can purchase our product and then the payment is automatically withdrawn from their account without them having to give us a check. As I understand it, they give a bank account number to the program and that is used as the source of their funds.

@seawulf575 touched on some of the related aspects of Fintech as an industry.

Essentially, Fintech is devising applications for conducting financial transactions from initial user to very end of transaction.

Example: You want to buy 100 shares of XYZ at $5 per share using broker Joe Broker.

Fintech would:

help you place the order.
Make sure you have funds to pay for it.
Route the order to an exchange.
Execute the order.
Take the money out of your account.
Record the transfer of ownership of the shares into your name.
Transfer the money to the seller.
Send you a trade confirmation.
Provide all the necessary transaction reports to Joe Broker for trade surveillance and compliance.
Do all that without any physical paper delivery.