The game is based on koikoi hanafuda, which is essentially Japanese poker. (Fun fact: the Nintendo company has been producing hanafuda cards since 1889.) The symbols on the cards are the
numbers kanji from 1-12, but it's not strictly necessary to know them. The cards at the bottom are your hand, the cards on top are the playing field and visible to you and your opponent.
The basic mechanic of the game is to take cards from the playing field in order to make sets, which are worth a certain amount of points depending on the set. You take cards from the field by matching them with cards from your hand. If you want a 3 card from the field, you must play a 3 card from your hand; if you want a 12 card from the field, you must play a 12 card from your hand, and so forth. (The deck consists of 4 sets of cards numbered 1-12 each.)
On your turn, two things happen:
1.You choose a card to play from your hand.
-If at least one card in your hand matches the number of a card on the field, you can capture. Both cards are added to your capture pile (on the bottom right) and automatically sorted into appropriate sets.
-If you can't capture, you must choose a card from your hand to add to the field.
-There's a special case when capturing, that happens if all four cards with the same number are visible to you in your hand and in the field. (Like if there are three 10 cards on the field and the fourth 10 card is in your hand.) In that case, you can capture all four in a single move.
2.After this, a card is drawn at random from the deck and played as if it were from your hand. If it matches number with a card on the field, you capture both. If it doesn't, the drawn card is added to the field.
It's then your opponent's turn, who goes through the same steps. The game ends when there are no more cards in play or in the deck, at which point the values of completed sets are added up.
I haven't played in a while, so the exact mechanics are kind of hazy, but IIRC, in Way, you get to have sex with your opponent as long as you have at least 1 point at the end of the match. However, the more points you have, the more the girl enjoys the sex, the more the girl's stats change in your favor*, and the more likely it is that the storyline will progress. The girl's score doesn't actually matter that much; you're not trying to beat her as much as you're trying to get a high score.
*: The player and the girl have stats, which change the number of cards on the field and in their hands. Basically, some girls are very hard in the beginning, but as you succeed in having nice satisfying sex with them they get easier and easier.
The actual sets you're trying to complete are a little unusual for someone who's not used to the cards since they're based on the imagery of the traditional deck, but you can see them all by right-clicking during a match. It's a little mystifying until you have the basics of it, so I'm gonna detail them here.
1st column
All these sets require the cards with the yellow numbers. The first two are self-explanatory (exactly 1-3-8-11-12 and 1-3-8-12). The third is any three of the cards on the left, plus the umbrella girl. The fourth is any three of the cards without the umbrella girl.
2nd column
The first is all three of the blue-bordered girl cards (6-9-10). The second is all three of the black-bordered animals (6-7-10). The third is 6 or more of the blue-, red- or orange-bordered cards in any combination. The fourth is 6 or more of the black-bordered cards in any combination.
3rd column
The first is all three of the orange-bordered girl cards (1-2-3). The second is the black-bordered 9 girl with the yellow-numbered 8 guy. The third is the same black 9 girl with the yellow 3 angel girl. The fourth is special and requires at least 10 of the common grey-bordered cards PLUS the black 9 girl; it's usually something you go for as a last resort if you have a shitty hand.
There are some cards that are obviously more useful than others (the black-bordered 9 girl, per example, is included in a bunch of easy combinations, as are yellow 8 guy and yellow 3 angel) so you usually want to go for them first if you can.
So that's about the basics of it. It's kind of a fun gambling game on its own merits, it has some pretty good erotic pics in it, and you get little dialog scenes which are actually funny and sometimes even touching, so if you can get the hang of the rules and the sets then it's all worth playing. Plus it teaches you hanafuda, in case you ever want to impress some Japanese people in a gambling den somewhere.
If anything's unclear, feel free to ask. I'm not super good at the game but I mostly have it figured out.