The basic idea of any karuta game is to be able to quickly determine which card out of an array of cards is required and then to grab the card before it is grabbed by an opponent. There are various types of cards which can be used to play karuta; it is even possible to create a version of this game which uses two standard poker decks for the cards!
I will use my made-up poker card version by way of explanation. I take one deck of cards and place them face-up on the floor. As for the other deck, I will shuffle it up and hold it in my hands. I will play the part of the reader; the other players will be the ones trying to grab the cards.
I turn over the first card in my shuffled deck. "Four of Spades", I announce. There is a commotion as the players scramble to get the Four of Spades. Once it is found, and the players return to their positions, I read the next card, "Seven of Hearts", and so on.
Later in the game, I start to say "Jack of Diamonds", but as soon as I say "Jack..." one of the players has already dove for the correct card: she knew that there was only one jack left.
The two "standard" types of karuta cards seem to be "uta-garuta" and "iroha-garuta".
In "uta-garuta", you try to find the last two lines of a tanka given the first three lines. It is often possible to identify a poem by its first one or two syllables. The poems for this game are taken from the Hyakunin Isshu.
Anyone who can read hiragana can play "iroha-garuta". Someone recites a proverb, and you go for the card on which is written the first syllable of the proverb along with a picture illustrating the proverb.