Chinese Ten
Introduction
In Chinese this game is known as Jiăn Hóng Diăn (pick up red spots) (揀紅點 or in simplified characters 拣红点). It is very similar to the Thai game Red Frog Black Frog (Gob Dum Gob Dang  กบดำกบแดง). The playing process whereby a card is flipped from the facedown deck after each play from a player's hand is characteristic of Oriental fishing games.
The description below was originally based with permission on the account by Jen Minute (Bullock) formerly available on the page http://members.xoom.com/_XMCM/niferbee/frames/Cards/chinese.htm which unfortunately disappeared around the end of 2001, along with the whole of her 'House of Cards' site. Additional information was contributed by David Halber, Stephen Coleman, Florent Barraco and Kantaphon Tongmee.
Players, Cards and Deal
The game is played with a standard 52card pack. The number of cards dealt to players varies depending on how many are playing. For a game with two players, each player gets twelve cards. For a game with three players, each player gets eight cards. For a game with four players, each player gets six cards. It is easy to remember how many get dealt to each player by simply dividing 24 by the number of people who are playing (i.e. 24 divided by 2 = 12, each player gets 12 cards).
After all the cards have been dealt, four cards are turned over face up from the remaining deck and placed in a layout like this:
X  X  

D  (X = card; D = deck)  
X  X 
The layout isn't that important, if you want your own setup, go right on ahead... (Just as long as you begin the game four cards out and the remaining deck on the table.)
Playing
The object of the play is to capture cards from the layout. A numeral card from Ace (1) to Nine captures another numeral card if their values add up to exactly ten  so for example Ace captures Nine, Nine captures Ace, Three captures Seven, a Five captures another Five, and so on. Tens and picture cards can only capture another card of the same rank: Ten captures Ten, Queen captures Queen, and so on. Suits don't matter for the purpose of capturing.
At your turn you play one card from your hand. If it captures a card from the layout, you take both cards and place them face down in front of you in your pile of won cards; if not you leave it face up as a new layout card. Whether you captured or not, you next flip the top card of the face down deck face up. If this captures a card from the layout you take both cards; if not it remains in the layout. After you have played one card and flipped one card, whether you captured or not, the turn passes to the next player.
Note that each card played or turned up can only capture one card from the layout. If the initial layout is 336Q and you play a Seven, you can only capture one of the Threes from the layout (3+7=10). If you play an Ace you do not capture anything (you cannot use 1+3+6=10 to capture two cards at once).
Note also that although the layout always begins with four cards, during the game there may often be more or fewer than four cards in the layout.
For another example, look at the following initial layout with the Jack of hearts, Ace of hearts, Six of clubs, and Two of hearts.
To make a move on a turn (with this layout), a player would need to play (out of his hand) a Jack (on the Jack), a Nine (on the Ace), a Four (on the Six), or an Eight (on the Two). Suppose he plays a Jack. He takes the pair of Jacks and stacks them face down in front of him (for later evaluation). He next flips over the top card from the deck and places it face up on the table. Suppose it's the Six of spades. This does not capture anything. It is now the next player's turn and the new layout is:
If, after capturing the Jack, the player had flipped a Four, capturing the Six from the table, then only two layout cards would remain for the beginning of the next player's turn:
If the rules are followed correctly, then you should finish with no cards in any player's hand, no cards in the facedown deck and no cards in the layout  on the final play, the last card from the deck will always capture the last card from the layout.
Special cases
If the initial layout contains three kings, three queens, three jacks, three tens or three fives, then the fourth card of that rank captures all three matching cards.
If the initial layout consists of four kings, four queens, four jacks, four tens or four fives, then the dealer automatically captures the four of a kind.
These two rules are necessary to avoid having cards in the layout that can never be captured.
Scoring
At the end of the game, each player should have a stack of cards next to him. When there are two players, only the red cards have a value; when there are three players, the Ace of spades also scores; when there are four players the red cards and both black Aces score. Black cards other than Aces are always worthless. The point values of the cards are as follows:
Cards  Point Value 

red 2  8  face value 
red 9  King  10 points each 
red Ace  20 points each 
Ace of spades  30 points when there are three or four players  otherwise zero 
Ace of clubs  40 points when there are four players  otherwise zero 
The tie score is
 105 points in a twoplayer game
 80 points in a threeplayer game
 70 points in a fourplayer game.
The difference of your score from the "tie score" is the amount you win or lose by. If your score is exactly the tie score you have neither won nor lost.

Example of scoring:
 Let's suppose a person had the following red cards in a twoplayer game:
 K, Q, 9, 6, 7, 3, 4, 5, 5, and an Ace of Hearts.
 The points would add up as follows:
 10 + 10 + 10 + 6 + 7 + 3 + 4 + 5 + 5 + 20 = 80.
Variations
Taiwan variant
Stephen Coleman describes a variation played in Taiwan. This is for four players. Deal and play are clockwise and the turn to deal passed to the left after each round (i.e. when all the cards have been played and scored). The dealer shuffles and the player to dealer's right cuts the deck and may look at the cut card, which will become the bottom card of the deck. The dealer plays first.
In this version the Ace of clubs is worth nothing; the Ace of spades is the only black card worth points (30). Red aces are 20 points, red K9 are 10 points, red 82 are face value as above. The tiescore (or 'base') is 60 points. It is often played as a gambling game: Losers pay into the pot $1 (Taiwan dollar) for each point less than 60, and winners collect from the pot $1 for each point over 60.
Red 5 Bonus. If a player manages to capture both red 5's in a single play, then the other players must each give him 10 of their points at the end of the round (by giving him one of their captured 10point cards). [The strategy to avoid this, if you have no red 5's in your hand, is to lay down a black 5 as soon as possible, hopefully forcing another player to play his red 5 on a black one, preventing the bonus.]
Florent Barraco describes a similar version for 2 or 4 players in which the red aces are worth only 10 points rather than 20, like the K9, red 82 are face value and the ace of spades is 30. This gives a total of 220 points, so the tiescore is 110 with 2 players and 55 with 4 players. Players pay or receive according to their difference from the tie score, and in addition a player who captures a red 5 with the other red 5 is paid 10 points by each of the other players.
Red Frog Black Frog (Gob Dum Gob Dang  กบดำกบแดง)
This Thai game, described to me by Kantaphon Tongmee, is played the same way as Chinese Ten, but the scoring is somewhat different.
 Red cards are always worthless.
 Black cards 29 are worth face value.
 Black 10JQK are worth 10 points each.
 The Ace of Clubs is worthless.
 The Ace of Spades is worth 50 points.