The bowers
The right bower is the jack of trump; the left bower is the other jack of the same color. Both count as trump, and they are the two highest cards in the hand.
The bowers for each trump suit
The left bower changes suit for the hand: the J♦ is a heart when hearts are trump, and it goes back to being a diamond on the next deal.
The order of trumps
Spades trump: the seven trumps, highest first. The J♣ has left the clubs; clubs are now A, K, Q, 10, 9.
Which jack is the left bower?
The jack of the other suit of the same color as trump. Hearts trump: the jack of diamonds. Diamonds trump: the jack of hearts. Spades trump: the jack of clubs. Clubs trump: the jack of spades.
Is the right or left bower higher?
The right bower is the highest card in the hand, the left bower the second highest. Then come the ace, king, queen, 10 and 9 of trump.
Does the left bower have to follow suit?
It follows trump, not its printed suit. When a trump is led and the left bower is your only trump, you must play it. When its printed suit is led, it cannot follow: play another card of that suit if you have one.
Can you lead the left bower?
Yes. Leading it is a trump lead, and the other players must follow with a trump if they have one.
What happens to the other two jacks?
They are ordinary cards in their own suits, ranking between the queen and the 10.
Why is a jack called a bower?
pagat.com: the word comes from the German Bauer, which means farmer or peasant and is also a word for the jack. The Semi-Official page spells it bauer too.
What are the most powerful cards in Euchre?
The right bower, then the left bower, then the ace of trump. A hand with both bowers and the ace of trump is sure of three tricks on its own.
In the app
Euchre March teaches the bowers as the second lesson, two hands on the real table with the coach, right after your first hand. The J♦ grays out when hearts are trump and diamonds are led, so you see the rule happen before you have to remember it.
Sources: pagat.com, Rank of Cards (“the Left counts for all purposes as belonging to the trump suit”); Bicycle, Rank of Cards; the Semi-Official Euchre Home Page, Bowers.