Firstly, of course, the cards only range from 9-A, leaving only five tricks per round and a heavy amount of voids. And secondly, the left bower jumps from one suit to another, leaving trump a slightly longer suit and the next suit slightly shorter.
Keeping these two facts in mind, while also considering the strategies below, is helpful for those who want to learn how to bid in Euchre.
Calling up the Card
The first question you have to answer in a round before you start thinking about any more complicated Euchre strategy, is "Should I make a bid and call up this card?" Obviously, you should only do so if you think your team can take at least three of the five tricks.
Keep in mind that the dealer gets the top card of the kitty, which is the trump suit. This presents a handy advantage if the dealer is on your team (since the dealer can gain a trump and often discard an unwanted card to create a void). Conversely, if the opposing team is dealing, calling up the card means a free trump for your opponent most likely to create an early void -- which can be trouble when you can only afford to lose two tricks.
Generally speaking, you can count on your partner for an average of one trick per hand in an average distribution. This means if you have two solid tricks, and your team is dealing, you should bid. If your two tricks are shaky, and the opposing team deals, you may want to pass.
Finally, as the dealing team, if a Jack is flipped up in the kitty, the dealer should almost certainly call trump and pick it up unless you have a terrible hand. A flipped jack is the right bower, and hence a guaranteed won trick. There's an old saying, "Turn down a bower, lose for an hour."
The exception to this is a no-trick hand. If you are the non-dealing partner, with a hand full of red cards when a blackjack is flipped, you may wish to pass the bid vehemently, so the dealer understands that you cannot be relied on for even a single trick.
Second Time Around
If all four players pass on calling up the top card of the kitty, it is turned face down, and players have a chance to declare any other suit as trump. If you have three of a suit (and ideally an off ace), this is a great time for you to declare trump.
The player left of the dealer should think hard before passing the opportunity to declare trump. Keep in mind that the dealing team has just turned down a free trump card because they were very weak in that suit. This means the dealing team is likely strong in some other suit, and if you give them a chance, they will make it trump.
Obviously, the best scenario is to be holding a pair of jacks of the same color. If you can call trump in either of those suits, you are guaranteed to win at least two tricks.
An Alternative Method
Most players will learn when to bid by feel, but for those who are not yet at that point or prefer something more concrete, you can assign a score to your hand, in order to determine whether you should bid.
Some players use a 3-2-1 point system, with the bowers worth 3 points each, face trumps worth 2 points each, and low trumps or off-suit aces worth 1 point each. In this system, any hand of 7 points or above is worth bidding.
Another system is simply scoring your hand with tricks. A right bower is worth a full trick. A Left bower is worth 3/4 of a trick. Ace, King, or Queen of trump is worth 1/2 a trick. And low trump is worth 1/4 of a trick. Each trump card in your hand beyond the second gets a 1/4 trick bonus. Meanwhile, off-suit aces are worth 3/4 of a trick but lose 1/4 of a trick for every other card you have in that suit. And finally, every suit in which you are void is worth 1/4 trick, if you have more than one trump card. Using this method, you can bid whenever your average hand value is at least 2.5 tricks.