Alternate scoring using cards
I grew up in Michigan, so inevitably I've been playing Euchre since the day I learned how to count to ten, and I must say that I've never seen anything other than the 5-5 method used for scoring. I'm not saying the others are necessarily wrong, but using the fives to keep score does have the advantage of being much more straightforward than the method lisa-ny describes above.
Here's how it works: First, find the 5s (all four of them). Since the fives aren't used during normal play, you'll find them in the discard pile that is set off to the side.
The fives in a standard deck of Western playing cards will usually look something like this:
+---------+ +---------+ +---------+ +---------+
|5 | |5 | |5 | |5 |
| S S | | C C | | H H | | D D |
| | | | | | | |
| S | | C | | H | | D |
| | | | | | | |
| S S | | C C | | H H | | D D |
| 5| | 5| | 5| | 5|
+---------+ +---------+ +---------+ +---------+
Note that there is a symbol in every corner, plus one in the center. (FYI, these symbols are called "pips")
One team gets the black 5s (clubs and spades), the other gets the reds (hearts and diamonds).
At the start of the game, both teams set one 5 face-up, and the other 5 face-down on top of it, so that no pips are visible.
When the first point is scored, the scoring team turns the top card 45° and slides it down slightly, so that the pip in the top-left corner is visible. To show two points, rotate the top card to horizontal, showing the two top-most pips. For three points, slide the card down to show the center pip. There are now 3 pips showing. And so on. Be careful not to bump the scorecards with your elbows, or you'll be accused of cheating. Also, don't use the scorecards as a coaster for your Faygo. That's just bad form.
When you get to five points, flip the top card over and lay it face-up on top of the bottom card (5 pips visible, 5 pips hidden). If you're in the lead, make sure the card makes a nice slapping noise when it hits the table. ;o)
This would also be a good time to announce gloatingly that you're "around the bend". (Others might say "around the horn" or "on the homestretch", depending on whether they're more nautically or equestrianly inclinded. A friend of mine always says "halfway to Mackinaw!", in reference to the great Port Huron to Mackinac sailing race, an annual tradition around here.)
For six points, slide the top card so that there's six pips showing (five hearts + a diamond underneath, for instance). And so on, until one team gets to 10 points and wins.
Euchre is a great casual card game because it looks complicated, but in actual fact is not. Once you gets the gist of things, there isn't a whole lot of thinking to do. There's a limited amount of "good plays" in any given situation. "Bad plays" are obvious almost immediately. There are some hands, called a "set-down loner", that are simply impossible to beat. You just say "loner in diamonds" and set the cards on the table, then add 4 points to your score. There's no betting, no math. Yah pick yr suit, yah play yr cards, yah count yr score, yah deal again. The main thrust of the game is just throwing the cards and finding out what everybody had. "Who's got the left? Huh, where'd the ace go? Musta been buried. Man, I had nothin'. Why'd you do that? I was three-suited. Damn you, I could have had a loner in spades!" etc etc etc.
In most cases the game is just there to give typically standoffish Michiganders an excuse to sit down and talk to each other, probably the real secret of its enduring popularity.