Vexed means "difficult and much debated." If your family is having trouble coming to an agreement about where to go on vacation next summer, your holiday trip has become a vexed issue.
When people can't resolve an issue or find a solution, it is a vexed problem — one that's become complicated because of differing and probably strong opinions. Vexed can also describe being irritated. If you borrowed your sister's car without asking permission, and if she had a fancy way of putting things, she might tell you she's incredibly vexed with you right now.