Doctors use terms that originate from Latin because it was considered to be a universal language of study. It helps to keep things consistent because different languages would translate terms into different things. This could cause confusion.
Hallux means your big toe. So both hallux valgus and hallux rigidus have something to do with your big toe. Hallux valgus is a bunion. A bunion happens when the joint of your big toe moves out of place, outward, and your big toe starts to turn in towards the other toes. As the lump grows, it can become more painful.
Hallux rigidus is a condition where the joint in the big toe begins to be stiff, or rigid. As the disorder worsens, it becomes more difficult to bend the toe as you would when you’re taking a step.