Hi squill. What you're experiencing is called 3D Touch. When you press firmly on an app icon, if that app supports 3D Touch, it will call up a menu of quick options. 3D Touch is also useful within many apps (you can "peek", and with more force, "pop" into a message in Messages from the messages list, for example).
As the other person said, if you want to move or delete apps without engaging 3D Touch, rest your finger gently on an app icon (it doesn't matter which one) for about two seconds. Once they begin jiggling, you can move or delete them normally as on the iPhone 6 and earlier. Press the Taptic home button to return to normal (just like the older iPhones).
I hope this helps!
-Pete