Chess needs no introduction: two players, sixteen pieces each, and a game deep enough that people have spent lifetimes on it without exhausting it. This browser version plays the full standard rules — castling, en passant, promotion, stalemate and threefold repetition — against a computer opponent at several strengths.
For a beginner the value is in the lower difficulties: an opponent that punishes real mistakes without being unbeatable is exactly how you learn. For everyone else it's a chess board that's always available.
Play often, and after each game look at where you lost material. Tactics — forks, pins, skewers — decide most games below expert level, and drilling them improves results faster than studying openings.
A move where the king moves two squares toward a rook and that rook jumps to the far side of it. It's legal only if neither piece has moved, the squares between are empty, and the king isn't moving into, through or out of check.
Yes — tap a piece then its destination. Nothing to install.