The Sultan (king) Pakubuwono II moved his court to the newly built Kasunanan Palace (also called Karaton Surakarta) in 1745. The palace was significantly expanded at the turn of the twentieth century by his descendant Pakubuwono X. However, many of the structures were destroyed by fire in 1985 and have been reconstructed.
The kraton is still home to the Sultan of Surakarta, which means that much of the residence is off limits. The palace is effectively cut into two large parts by a somewhat busy roadway. One part is the 'outer' public spaces of the court, with a huge pendopo (open-air pavilion) for dance and musical performances. Behind the big pavilion is a smaller pendopo for private audiences with the Sultan. Since there's a separate admission fee for this area, I'd suggest skipping it in favor of the main palace.
The second, more interesting part of the palace still houses the living quarters for the Sultan's family. The palace buildings are arranged around several large courtyards. One of these has been converted to a museum housing items once used within the palace, such as old horse carriages, palanquins and even some statues from area temples such as the Prambanan. Next to the museum is another courtyard, filled with sand and at the center of which is a large pavilion still used by the royal family for audiences and private performances. While you can look at the pavilion, entry is forbidden. At one end of the courtyard is a lighthouse-like tower that is visible from some distance around Solo.
Admission to the front palace area is 5,000 Rupiah (0.30 USD). For the main palace and museum, the admission is 15,000 Rupiah (0.90 USD).