What can the Tesla app do?

Aside from scheduling service center appointments or otherwise communicating with Tesla Service, the Tesla app (for either iOS or Android) is one of the primary ways you can control your car. It should have already been set up as your Phone Key. You can control some aspects of charging (stopping, starting, charge level), you can turn on climate control and change its settings, you can lock, unlock, vent windows, open (and close if you have the appropriate hardware) frunk and trunk, turn on or off Sentry, Summon your car (depending on what features you purchased, that includes Smart Summon), you can request immediate Roadside Assistance, and you can of course spend more money by buying available upgrades.

There are lots of third party applications as well, which can provide handy analysis of battery usage, battery health, trip and charging statistics, and even provide some automation including voice commands to do all of the above. Note that third party applications do require you to authenticate with your Tesla account username and password and get a token, which some people are not comfortable with.