Plaza Mayor is a very historic site in Lima; it is actually the place where the Spanish conquistador Francisco Pizarro founded Lima in 1535. Despite the fact that there are very few monuments dating back to those times, the brass fountain is one of the oldest relics that you can see here. Nowadays you can admire its Colonial Baroque buildings, like Lima's Cathedral, the Archbishopric Palace, the Government's Palace or the Lima Municipality.
Plaza Mayor de Lima — Photo courtesy of J. Enrique Molina
Take note that not only was Lima founded here, but the Peruvian independence was first officially proclaimed here in 1821. 
Brass Fountain Plaza Mayor — Photo courtesy of J. Enrique Molina
This is also a good point to start your urban exploration of Lima, since it's located nearby other historic places worth visiting, such as the San Francisco Church and Convent, Iglesia de San Pedro, the Plaza San Martin, the Jiron de la Union, the Barrio Chino (China Town) and some traditional manor houses like Palacio de Torre Tagle (current headquarters for the Chancellory).
Or you may also want to finish your morning of sightseeing here. It's a good idea, since you can try having lunch at T’anta Centro Historico or at Paseo de los Escribanos (beside Lima Municipality). There are plenty of good restaurants around, including a vegetarian one.
Once you’re ready to wind down and relax, take a ride on a calesa (horse-pulled carriage) to enjoy a slow, leisurely ride around Plaza Mayor. If you still have time for more action, take the small buses running from here to the top of Cerro San Cristobal, but take note that these are only in service on weekends. 
Calesas — Photo courtesy of Wikimedia Commons
There is always something different to see here, especially popular fairs and folk dance events, usually scheduled for weekends.