The center of town in Copán Ruinas [1] is, of course, the parque central, which was redone a few years back with a good dose of concrete. The semicircle of tile roofing across the center and the castle-like wall on one side look rather out of place to many foreigners, but the locals love it.
The municipal market has just moved into a beautifully restored colonial-style building half a block east of the square; the piles of fresh fruits and vegetables always making for an interesting browse.
In addition to viewing the Mayan ruins, many hikes and excursions can be made in the hills and valleys around Copán Ruinas. The countryside is generally very safe for wandering, and local campesinos are helpful to visitors who lose their way. One can also stroll along the Río Copán in any direction from town and enjoy the rural beauty of the valley.
Note: Several tourists have been robbed walking up to El Rubí waterfall near Santa Rita [2]—since then, many tour companies send visitors out with an armed guard, which is hardly relaxing. Ask at your hotel or one of the guide companies for more information if you are considering this hike.
Links:
[1] http://www.moon.com/destinations/honduras/western-honduras/copan-ruinas
[2] http://www.moon.com/destinations/honduras/western-honduras/copan-ruinas/sights/santa-rita