World Tour is probably where you're going to spend most of your time — this is the core of Super Mario Run.
In most levels of World Tour, your goal is to get from the start of the level at the left to the end of the level on the right, where you'll find the iconic flagpole. You'll jump over obstacles, enemies and gaps and collect coins along the way.
If you want to do everything there is to do, you'll also be replaying every level three times (at minimum). Mixed in among the standard gold coins in each level are five pink Challenge Coins. Collect these to unlock the purple Challenge Coins. These purple coins will be scattered through the same level in different and harder-to-reach locations or inside blocks. Collecting all five of these will unlock the black Challenge Coins. These are usually even harder to get to. Think of collecting Challenge Coins as getting three stars in an Angry Birds level.
Just one more thing: Completing these levels also earns you Rally Tickets that you can cash in to play the Toad Rally mode.