With only a couple of days in Zion, we managed to fit in several hikes:

Angels Landing: One of the most popular trails in the park. If you're in Zion and only have time for hike, then this is the one to take
Observation Point: Much higher and longer (and therefore quieter) than the Angels Landing trail, this hike is surprising because it is even more scenic.
Hidden Canyon: Branching off the main trail that leads to Observation Point is a short side-trip into Hidden Canyon.
Weeping Rock: Just a short hike from the valley floor
Emerald Pools: Why not call them the algae pools, because that's all they are.
Riverside Walk: At the top of the canyon, where it narrows and the road ends, a hiking path can be followed further up the canyon. When the paved path ends, it's possible to continue several miles up the canyon by hiking up the river, if the water level is low enough.
Canyon Overlook: Near the exit to the park is a short trail that is well worth a look.