This is one of our favorite card games. We don't do a lot of card games because Hannah can get very competitive which leads to tears and tempers flaring and general pouting. And that's just me. Her behavior can get hairy also. So we add certain options that make for more cooperative play (like sharing a card that you don't need but you know someone else needs).
The game is called Garbage and it's a game that is easy to learn and fun to play. We use really big cards, seven inches tall, that I got on clearance years ago. They're made by Fundex. They're easy for the young ones to hold and see.
Players: 1 - 4. This is a good solitaire type game for kids. If you want to play with more than four people, use two decks. Ages 3 + (depending on how much you're willing to help).
Setup: Dealer deals eight cards to each player. Remaining cards form a pile between all players. Players do not look at their cards but place them face-down in front of them in two rows of four. Players mentally number each card beginning with '1' and ending with '8'. (We use numbers written on little pieces of paper above each card until the child is able to remember what number she's assigned to each card.)
Play: First player draws from the top of the extra card pile. If the card drawn is any number from 1-8, the player places it face-up in the correct place and looks at the card it replaced. If that card can be used to replace any of the other numbers, use it (and keep going until you get a card you can't use) - if not, place it on the discard pile. (For example, player draws a '2' card and puts it down in the '2' spot. They then look at the card that was face-down in that spot, which is a '7'. They put it in the '7' spot and look at the card they replaced from that spot. It is a '10', so they put that in the discard pile.) The next player can use the discarded card or pick a new card.
When a player has replaced all eight cards, the round ends. All players turn in their cards and the next round starts. In this round, the player who won the last round only has seven cards to replace while everyone else does eight cards again (this is obviously another place where we bend the rules a bit - when we play, all players go down a number instead of just the winner of the last round). After this round is over, it goes down to six cards and so on until there's only one card left - the first player in that round to get an 'ace' wins the game.
I've mentioned before that math, as a subject, is a difficult one for me to feel confident helping my children with. Games like this really help the child learn number concepts but keep math from being separated from life (and into a subject) and help keep me relaxed about something that is far too easy for me to get wound up about.