Two classical games drawn, two rapid games drawn, two blitz games drawn, and somebody has to fly home tonight. So the organisers stop trying to be fair and play one last game that is deliberately lopsided.
Time on one side, draw odds on the other
White gets more time on the clock. Black gets draw odds, which means a draw counts as a win for Black. There is no such thing as a level result, so the match ends when the game does.
A split of 5 minutes against 4 is the most common one, and many regulations add a small increment from about move 60 so that the last phase is played rather than flagged. Everything else is ordinary chess. Black still loses on time, still loses to a checkmate, and still has to make legal moves in a scramble.
The bargain is meant to be close to even: White has the clock but must produce a win, Black can head for a draw but has to do it with less time. Strong players disagree about which side they would take, which is the best evidence anyone has that the trade is roughly fair.
How the colours get decided
Two systems are in use. The simple one is a drawing of lots, with the winner choosing a colour. The clever one is a bidding auction: each player writes down, in secret, the lowest clock time they would accept in order to play Black, and the lower bid gets Black with exactly that time. It sets the price of draw odds by making the players name it themselves, and it produces some very fine judgement calls at the low end.
Where you will see one
The FIDE World Cup finishes its knockout matches this way when the rapid and blitz tiebreaks have not separated two players. Norway Chess plays one after every drawn classical game, so a handshake there is only the start of the afternoon. World championship match regulations have provided for a final Armageddon game for years, and no match has ever needed it: the rapid tiebreaks have always settled things first.
Club chess borrows it too, usually for a league playoff or the last board of a match that has to produce a winner before the hall closes.
How the two sides actually play it
White must attack. No queen trades, no symmetrical lines, nothing that heads for an ending. A dead level rook endgame is a loss, and every simplification is a small step towards defeat.
Black should be trading from move one. Every exchange helps, a locked pawn structure is a friend, and a position where nothing can happen is already a win.
7 
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c The other half is clock management, and it is where the extra minute usually goes to waste. White thinking for ninety seconds on move twelve has given the whole advantage back, and Black playing a solid line at speed is the plan working exactly as intended. If you ever get handed one of these, spend an evening first on rook endings: holding a level one, or a level one a pawn down, is the single most useful skill either colour can bring to the board.