Constructed in the 16th century in Renaissance style, the Museum of the Royal Houses (Calle de las Damas, tel. 809/682-4202, 9 a.m.–5 p.m. daily, RD$30) was the Palace of the Governor and the Audiencia Real (Royal Court), which was a panel of judges created to check the power of the governor. Just outside and opposite the museum a large sundial was positioned so that those inside could simply glance out the window and tell the time.
The Museo de las Casas Reales has an excellent re-creation of the colonial era using real colonial-period objects, including treasures recovered from sunken Spanish galleons, gilded furniture, art, and replicas of the Niña, the Pinta, and the Santa María. Each room has been re-created according to its original decor and usage. Taíno artifacts are on display as well.
An impressive and varied collection of weaponry that was donated by Trujillo is on display on the second floor. If you only have time for one museum in the Ciudad Colonial [1], make this one it.
Links:
[1] http://www.moon.com/destinations/dominican-republic/santo-domingo/sights/ciudad-colonial