I am looking for an old cartoon—probably made around the '80s or '90s—that was about people who had different colored wristbands that gave them animal-like powers. I think it used to be shown on Boomerang.
Although the time period when it was made does not match what the questioner thought, I think the show he is looking for probably is, in fact, Legend of the Dragon. The show aired from 2005–2008, and it was not actually on Boomerang either; it was broadcast in America on Disney-owned channels and on other channels elsewhere in the world. (I looked at the long list of shows broadcast on Boomerang, and I did not see anything more promising that was older—although it is certainly conceivable that I overlooked something.)
The show starts when the previous Golden Dragon has passed on to the afterlife and Ang and Ling Leung, seventeen-year-old fraternal twins, born in the year of the Dragon, are next in line for the Golden Dragon power band and user of its mystical abilities to defend the mortal world from evil. Ang has no belief that he will become the Golden Dragon, as Ling is the best fighter at the Dragon dojo. Ling is positive that she will become the Golden Dragon. However, when Master Chin calls upon the power of the Golden Dragon to choose who will be the next golden Dragon, it chooses Ang. Humiliated and furious, Ling storms out of the temple and is tempted by the ultimate dark and evil power of the shadow Dragon. From then on, Ang is forced to fight his own twin sister.
The second and final season (which consisted of just thirteen episodes) begins with the reformed Ling finally back home as the second Golden Dragon alongside her twin brother. Now they, Chin, and the fellow Guardians of the Tiger and Monkey Temples are ready to put an end to the Zodiac Master and the Emperor of the Darkest Ying once and for all. However, the Emperor is on the move to regroup at the Temple of the Shadow Dragon, but not before he attacks and disintegrates the Temple of the Golden Dragons, leaving Ang and Ling's power-bands on the verge of powerlessness. Luckily, Chin suggests going to long-lost Temple of the Shadow Dragon and utilize its vast amounts of negative energy to rejuvenate their power-bands. As Ling was exposed to the evil energies of the Shadow Dragon for quite some time, she becomes easily tempted by the temple's negative influence on both her mind and spirit. However, with the wisdom of Master Chin, she and Ang successfully recharge their power-bands with no negative effects. Elsewhere, the Emperor manages to release his younger sister Yin Wi, the Guardian of the Shadow Rat, who had been imprisoned by one of the previous Golden Dragons for one thousand years. The two siblings concoct plans to conquer the modern world, and have their vengeance on all twelve guardians of yang; the Golden Dragon in particular. As they head to the Temple of the Shadow Wolf, the heroes get there first and encounter K-Lo, the Shadow Wolf Guardian who was one of Ling's allies during her dark times as the Shadow Dragon and who seems to have quite an interest in her, now that she has become a Golden Dragon.
The characters' dragon or animal powers are utilized through the power-bands mentioned above. There were twelve sets of powers, based on the years of the Chinese Zodiac: rat, ox, tiger, rabbit, dragon, snake, horse, goat, monkey, rooster, dog (interpreted as wolf on the show), pig. The most prominent villain was the Zodiac Master. Each band had a color, indicating its elemental affiliation,
and a gem carved with a representation of its tutelary animal.
Activating the power-bands allowed the guardians to take on animal-empowered forms.