Communicating to your partner what suit to lead back is an essential skill in euchre. Good players don't just randomly select the cards they play, but instead, read their partner's signals. Not hand signals, but a message sent by the partner's choice of the card (or suit) played. Armed with this information the partner can lead back the correct card or suit. You can give your partner a lot of information simply by the way you play your hand. All the top euchre players work as a team, with each member watching for the visual card clues given by their partner.
One classic situation is leading the right bower and finding your partner void in trump, but they hold an off-suit ace-king combination. By playing the ace on your lead, they are signaling that they hold the king of that suit. You now know that leading that suit back would be the best way to bypass the opposition and give the lead to your partner. Of course, this also tells the opposition what not to lead, but it will help your team more than it will help them. Keep in mind that this works not only on a trump lead but on any lead where a partner cannot follow suit.
Often your partner won't have an ace-king combination but instead will have an ace along with a lower card of the same suit. In these cases, your partner would play the lower card indicating they have the ace. While this is not as straightforward or definitive as showing an ace, it's a good indicator and better than nothing. (Note that even if they are throwing off a singleton thus creating a void, it is still a good suit to lead back, as your partner will be able trump in).
-> It is important to note here that the rules change if your partner is the dealer. Why? Because the dealer already had the opportunity to short-suit his hand, thus creating a void. Therefore, when they throw off, they are likely throwing from a suit in which they hold two cards. Say hearts is trump. You lead the ace of diamonds and your partner (as the dealer) throws off a club. The best way to get back to them is to play a spade, not a club! They most likely will be void in spades, so leading that suit will give them an opportunity to trump in and take the trick.
In this example, the dealer sits in the South position and has just turned up the 9 of diamonds. No one orders and the dealer picks the queen.
First seat(W) starts off by leading the ace of hearts. This plays into Norths hand perfectly and allows them to use the ace of trump. East follows suit with the queen. The maker(S) is void in hearts so they throw off an unneeded jack of clubs. One trump played, with the maker's team having their first trick.
Now the question becomes, what suit to lead back that would the maker be able to take?
Taking a cue from his partner's discard, North seat leads back the 9 of spades. Note that, had second seat lead back a club, third seat would have made their ace of clubs good, costing the bidding team a trick and possibly even a euchre.
As you can see the dealer already holds two trump. The pick-up gives him a third, making a hand strong enough to call on. While it is possible the dealer could have started with a set of three, it's more likely one off-suits will be a singleton. Understanding this allows a partner to bypass the left-hand opponent and lead a suit the dealer can trump.