A wry sense of humor is a sarcastic one. You were late for work, stepped into a mud puddle, and you forgot your lunch. If your co-worker asks how your morning is going, you can reply with a wry tone, "Perfectly perfect."
The original use for the adjective wry was to describe something that was bent or twisted, so a sprained ankle could be described as "a wry ankle." Today, the word wry is often used to describe less tangible twists. Wry humor and wry wit both describe a sense of humor that is a little twisted from the norm.