Whenever you're too busy to take care of your pet - for example, you need to go to school, or to work, where you cannot play with your Tamagotchi - you can leave it for a couple of hours with the babysitter.
The babysitter will take care of your character from 7 am to 5 pm. During this time your character will not get hungry, unhappy, or sick - so it is basically a pause feature.
To leave your character with the babysitter, select the clover icon, and then the 5th option. The babysitter will come, and then a question if you want to leave your Tamagotchi with the babysitter will appear on the screen. The first option means 'yes', the second one means 'no'.