You can run Bitcoin Core with the walletnotify feature, which will execute a script of your choice every time you receive a payment.

See How do I use walletnotify?

There are also lots of third-party service providers which have APIs to notify you of incoming payments.

Wallet Notify works great, however, its worth mentioning the advantage of using a 3rd-party service over Bitcoin Core. You have the ability to track ANY address, wherever it may be located (local pc, cloud service, mobile device, etc). You can even track addresses that you don't own (clients, partners, competitors, etc).

If you're looking for a enterprise solution, BlockCypher's API offers a comprehensive suite of popular coins and services.

BitNotify offers a quick and simple email notification service.