Recovery mode is a useful way of fixing an iPhone (or iPad, for that matter) that cannot be recognised by iTunes, or is stuck on the 'Apple logo' screen. It's slightly different to DFU mode, which is more drastic and used for tinkering with firmware and the like; recovery mode just attempts to do a clean install of the latest version of iOS without erasing all your data.
Here's how to put your iPhone into recovery mode.
Check that iTunes has been updated to the latest version, and then close it down.
Now plug your iPhone into your PC or Mac using the USB cable, and open iTunes.
Press and hold the power and Home buttons at the same time. You'll see the Apple logo after a while, but carry on holding down the buttons; wait until you see the recovery mode screen.
(The iPhone 7 or 7 Plus have non-mechanical Home buttons that don't work when the devices are powered off. If you've got one of these, you'll need to press and hold the power and volume down buttons instead. Keep them held until you see the recovery mode screen.)
You will be presented with the option to Restore or Update - you should choose Update. If it is able to, iTunes will install the latest version of iOS but retain all of your data.