If you've tried to rotate your phone "upside-down" (charging port towards the ceiling) and your screen doesn't rotate, chances are your phone has the "orientation lock" enabled. Read on to find out how to turn it off!
First, swipe up from the bottom of your screen.
This will bring up a sheet called the "Control Center" on your iPhone. On it, you will find an icon that looks like a padlock enclosed in a circular arrow (this is circled in white below.) It may not be in the same place as the image depending on your phone.
If this icon is not greyed out (it may be white with a red background like the image above, or red with a white background on some phones), then orientation lock is enabled and your phone will not rotate. Tap on this icon to turn it off.
If you have successfully disabled orientation lock, your phone will display a message like the one seen below, either at the top of the control center or the top of your screen. You will also see that the icon becomes "greyed out."