Update: You can disable bubble and screen effects in iOS 10 and later by navigating to Settings > General > Accessibility and switching on the Reduce Motion slider. This will disable all message effects on your device.
iOS 10 or later allows iOS users to send other iOS users animated effects through the Messages app. These effects can be individually selected or triggered by typing certain phrases. For example, sending a text message the reads "Happy Birthday" will trigger an animated balloon effect on any iOS devices running iOS 10 or later. As of right now there doesn't seem to be any way to disable this feature in the Settings app. You can see a list of phrases that trigger the animations here so you can avoid accidently sending them in the future.