And indeed, it turns out that every player has a diamond card. So when a diamond card is played, the ace of diamonds in our opponent's hand takes that trick.
If you keep following along with the replayer, you will see that we euchred our opponents, despite this tactical mistake on the part of my teammate! We had some luck: we both had two trump cards, and I had the strongest one, the right bower (jack of hearts). In addition, one of our opponents had a pretty bad hand, with no aces and no trump cards.
It definitely takes some luck to win consistently at Euchre! However, sometimes, you can "make your own luck" by using a good strategy. Remember to "go for the void" when you have the opportunity to discard a card.