Unified Payments Interface. It is an instant payment system developed by the National Payments Corporation of India. You need an Indian bank account or debit (ATM) card and a smartphone (some feature phones can also use UPI). You install any one of the UPI apps, and bind your account / card to it. You can send money to someone by entering their mobile number, UPI ID (any unique alphanumeric string that isn't already taken; this is useful if you want to hide your number) or QR code (most shops will display this at the entrance). The NPCI sets the standard and maintains the backend; users can use any standard-compliant app. It is also okay if the recipient uses a different app from the sender.
There is no fees for personal transactions, and business transactions up to Rs. 2,000. Business transactions above that will be charged 0.4% starting tomorrow, but this is still lower than what most cards charge.
Outside of India, UPI is interoperable with the national payment systems of Singapore, Malaysia, Oman and Qatar, and with some banks in the UAE.