The Local Timeline
The Local Timeline only shows public posts made by users on your home instance. This can be useful if your instance has particular community norms that users on other instances may not have, such as particular topics that get put under content warnings; or particular in-jokes and shared interests.
To view the Local Timeline, click the "Local Timeline" in the Compose column or the respective button on the Getting Started panel.
How exactly is Mastodon decentralized?
There are different ways in which something can be decentralized; in this case, Mastodon is the "federated" kind. Think e-mail, not BitTorrent. There are different servers (instances), users have an account on one of them, but can interact and follow each other regardless of where their account is.
You can use Mastodon with any of these apps!
Find your Twitter friends on Mastodon