Hi,
Which exact HP printer model are you using?
The AirPrint technology works with wireless (Wi-Fi) or wired (Ethernet) network connections. This means that you would connect an AirPrint printer wirelessly or by Ethernet to the same Wi-Fi router that is used for a wireless connection from an iPhone. USB connections are not supported.
AirPrint as such does not require an additional app. The printer manufacturer may provide a separate app for setup and ink management purposes, though.
If your printer is an AirPrint-enabled model, and there is a problem with device discovery, you could try the following procedure (in this order, with these waiting times):
Power off the Wi-Fi router, the HP printer, the iPhone, and all other devices/computers. Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the HP printer. Wait a minute or two. Power on the iPhone. Finally, power on all other devices/computers.
About AirPrint - Apple Support
Use AirPrint to print from your iPhone, iPad, or iPod touch - Apple Support