Engagement jewelry is a category of rings and amulets that is exorbitantly more expensive than regular jewelry. It will raise a pet or hired adventurer's affection if given to them. However, pets prefer to wear engagement jewelry rather than most other equipment, and react badly if one tries to take the jewelry back (they will angrily swallow the item, thus destroying it, and your relationship level will take a big hit).
If you are going to give your allies engagement jewelry, be warned that they are limited to 20 inventory slots; since you cannot remove the jewelry without taking a big relationship hit, it is advised that you only give them rings/amulets that can stack (with the same attributes, material type, and blessed/cursed status). Beware - if the rings are equipped, they will not stack. This will reduce the amount of space that is permanently sacrificed for the relationship gain. If you are just trying to get the pet's affection high enough to marry and your pet is low on inventory space, taking the jewelry away afterward may be the best option. In general, from a baseline of 0 affection, it takes about 15 pieces of engagement jewelry to reach the Soul Mate status. You can also avoid the whole inventory issue by waiting until the pet is already up to Fellow status (keep taking them through the Puppy Cave and let them get all the kills), and just give a few more engagement items to get it to Soul Mate, then take the items back after marrying them.
Other than the higher price and affection gains, it acts as regular jewelry. Because of its high value-to-weight ratio, it can be a good thing to trade with other adventurers.
Moyer the Crooked is the best source of purchasable engagement jewelry, but he sells at gouged prices.
TIP: A good way to farm engagement jewelry is hunting repeatedly at the bottom-most level of the Puppy Cave (Floor 4). Wizards of Elea spawn frequently there, and are generated with two rings and a necklace, all of which might be engagement jewelry. Killing them may net the jewelry as a drop, but a good Pickpocket can speed up the task (just remember that stolen items gain the "It's a stolen item" attribute, thus have their value reduced by half).
Elona+ gave unique sprites to engagement jewelry and also gave innate attributes to certain types of equipment and materials. In the case of engagement jewelry, rings now grant an extra +5 Charisma when equipped, while necklaces grant an extra +5 Willpower. Other than that, they function the same way as in the original game.
A nice trick to increase Charisma and Willpower quickly is to use (cursed) scrolls of inferior material on jewelry you don't need until they come out as made of raw/candy, then eating them to train the corresponding stat.