A troth is a serious promise to be faithful, such as a pledge people make about getting married. If you and your girlfriend announce your troth at a family dinner, then that means you better be serious about your relationship.
You can pronounce troth to rhyme with "cloth" or to rhyme with "oath." In fact, oath is a synonym for troth. Both words are related to truth, in that when someone makes a troth or an oath, what is said is taken as the truth. You can see troth in words like betrothal, which means engagement, or betrothed, which refers to a person someone is engaged to.