Most modern printers work with phones without a computer in between. iPhones use AirPrint, Android phones use the built-in print service or the printer maker's app, and both need the phone and the printer on the same Wi-Fi network.
Why can't my phone find my printer? Usually the phone and the printer are on different Wi-Fi networks, such as a guest network or different bands. Put both on the same network, and check the printer is awake and not in a deep sleep mode.
Can I print without Wi-Fi? Yes, with Wi-Fi Direct if the printer supports it, which connects the phone straight to the printer. Some Android phones can also print over a USB cable with an adapter.
Do I need to install an app to print from my phone? Not for most modern printers. AirPrint and the Android print service are built in. The maker's app helps with older printers and adds extra features such as scanning.