Something that's inelegant isn't polished or cultured, but instead is awkward, rude, or ungraceful. Your loud footsteps clomping down the hall of a palace as you race to meet the Queen are inelegant.
Your inelegant tap dance probably won't win first prize in the school talent show, and your brother's inelegant behavior might get him thrown out of an extremely formal party. In both cases, the required amount of grace just isn't present — in your brother's case, it might be his inelegant laughter or his inelegant clothing that cause him trouble. The Latin root, inelegantem, means "without taste" or "without judgment."