Hex is not hexagonal chess. The board is a diamond of hexagons, and the whole point is to connect your two sides with one unbroken chain. The guide below follows the order of a real first game. Read it once, then open the AI board and start with Black left to right and White top to bottom.
If You Remember Only Three Things
First, Black connects left to right and White connects top to bottom. Second, stones do not move after placement and there are no captures. Third, the first player to make one unbroken connection between their two sides wins. If you remember those three points, you can already start your first game.
How to place a stone
Black plays first. After that, players alternate one stone at a time on an empty hex cell. Stones never move after placement, and there are no captures. This site uses a fixed 11×11 board. In AI play you can choose Black or White; in local two-player and online rooms, players simply alternate turns.

Which sides belong to whom
The two dark borders on the left and right belong to Black. The two light borders at the top and bottom belong to White. Stones of the same colour connect when they share a side; on a hex board, even cells that look diagonal still count as neighbours. Black wins by connecting left to right. White wins by connecting top to bottom.

The second player may swap
A first stone near the centre usually helps both colours, so after seeing it the second player may swap, take that stone as their own, and leave the opening player as White. On this site, AI games show the swap control when it is legal.
How you win
You win by making one unbroken connection between your two sides. The chain does not have to be straight; if your stones link side to side all the way across, it counts. On a full Hex board, a draw is not expected, and this site ends the game as soon as one side connects.

What changes by mode
AI games include hints, undo, replay, and the swap rule, which makes them the easiest place to learn. Pass-and-play uses one device and gives no move hints to either side. Online rooms sync accepted moves and do not offer hints or undo.
Questions
Is Hex the same as hexagonal chess?
No. Hex is a connection game. Hexagonal chess is a chess variant with pieces, and this site does not play that game.
Do diagonal-looking cells count as neighbours?
Yes. Hex cells have six neighbouring directions. If two cells share a side, they connect; the board is not limited to up, down, left, and right.
Can I pass my turn?
Not in ordinary play. On this site, the only special option is swap in AI games after the first stone, when it becomes the second player's turn.
Why is the first move often not placed in the exact centre?
A centre-adjacent opening helps both colours. The second player will often swap to take it, which is why many openings begin slightly off-centre.