The Charleston in Mahjong Island works as a tile swap before play: every player passes 3 tiles right, then 3 across, then 3 left. "Across" means the player sitting opposite you. If every player votes yes, a second Charleston passes 3 left, then 3 across, then 3 right.
When does the Charleston happen?
The Charleston happens right after the deal and before anyone takes a turn. Each player is dealt 13 tiles, and the dealer gets a 14th tile. The dealer discards first, but only once the Charleston is done.
What does a pass look like?
You pick tiles to send away and keep the ones you want. Say your hand holds two Red Dragons. That is a Pair, two identical tiles. You might keep them and pass 3 other tiles instead.
Why pass tiles at all?
The Charleston gives you a chance to trade tiles before play starts. You win by completing one hand on the Island Card: 14 tiles that match one line on the card. You can pass tiles that do not fit the line you want.
To see how the rest of a hand plays out, read the Mahjong Island rules for winning a hand. To pick a line to aim for, browse every winning hand with examples.