Euchre is perhaps a unique card game in that cheating is considered a normal and legal part of play in many circles. While obviously, you should never cheat against people likely to take it badly, there are many cases where cheating is perfectly acceptable among the group you will be playing with (usually only when you're not playing for money). If this is the case, here are the top three accepted methods to cheat at Euchre.
Steal the Deal
This is hands-down the most common Euchre cheat in use, probably because it requires no skill or practice to pull off. Simply put, since the dealing team gets to pick up an extra trump card, it is always an advantage to have your team deal. According to the standard rules, the deal rotates clockwise around the table. Stealing the deal is very easy; just gather the cards and begin dealing when it's technically the other team's turn to deal.
If you finish dealing out the cards before anyone points out that it isn't your turn to deal, then you have successfully stolen the deal. Since cards are dealt in two rounds around the table, rather than five, this is not difficult to do.
When stealing the deal, you want to deal fast enough that nobody realizes what has happened until it is too late, but not so fast as to call attention to yourself. If you just dealt, sometimes it is better to gather the cards and pass them to your partner, because the same person dealing twice in a row will be more obvious.
In addition to gaining your team an extra trump card, the advantage of being the dealer is that most other cheat methods rely on your access to the cards, which is a lot more difficult if you are not the dealer.
A time-honored cheat by the dealer is to sneakily deal an extra card to yourself or your partner, leaving only three in the kitty. While dealing an extra card might be obvious in most card games, Euchre's distinctive dealing tradition dictates that all cards be dealt out after two rounds of the table, generally in clumps of twos and threes. This makes it easier to conceal an extra card by either lining it up with another so three cards look like two, or simply by dealing three cards both times around the table if people are not paying attention.
If there is any worry about potential cheating, a dealer is always under the most suspicion. For this reason, many dealers using the six card trick prefer to deal the six-card hand to a partner, rather than themselves. If the opposing team is carefully watching the dealer's cards, they will still only see five cards, because the actual cheat occurred across the table.
For this to work, though, it is essential that your partner does not ruin your cheat by acting befuddled by the extra card, or obviously fanning six cards. The sixth card must be disposed of secretly, usually in the middle of a trick you or your partner will collect, otherwise it will be stuck in your hand at the end of the round and make your cheating obvious.
If your opponents are not familiar with this cheat, you can warn your partner by talking about a trip to Six Flags. If your opponents are wise, you are best served to avoid table talk. Regardless, the dealer must take care to conceal the number of cards remaining in the kitty, never letting them become spread out lest the three cards (rather than four) make themselves obvious.
A slight variation on the extra card trick, the Double Draw requires the dealing team to have the card called up. As the dealer, when you go to pick up the trump card, you instead grip it along with the card directly under it. Ideally, by keeping the cards together, you can avoid notice. You now have an extra card in your hand, seven instead of six.
Like the Six Flags trick above, the extra card gives you options, but needs to be disposed of before the end of the round, which relies on your being able to hide it in a trick that your team is picking up.
Rather than risk getting stuck with an extra obvious card in your hand, when you (as the dealer) discard your card face-down atop the kitty, you can simply discard two cards held together as one, just like you picked them up. This keeps your hand at five cards, so nothing tricky will be required during the round, and the double discard makes it easier not only to stock up on trump, but also to create a void.
Obviously, any standard card deck cheat (e.g. stacking or marking the deck, dealing from the bottom, peeking at cards, etc.) might function as well in Euchre as anywhere else, but while the Euchre-specific cheats above are often accepted as "amusing quirks" of the Euchre scene, nobody likes someone who reneges on a trump, deals from the bottom of the deck, or other similar cheats. That is simply not how to cheat at Euchre.