After a stint as a USAF military airfield during WWII, Ciampino became Rome's main airport until 1960, when Lionardo da Vinci Fiumicino Airport was opened. In 2007 the terminal facilities were expanded for the new class of low-cost airlines.
Today the airport hosts Ryanair and Wizz Air, a Hungarian carrier with connections to Eastern Europe.
The two markers on the map show the locations of the airport and the train station.
As you can see from the map, the airport is conveniently located about half way between the center of Rome and the volcano that holds Lago Albano and the Castelli Romani, a favorite area for summer tourists (and popes) aiming to escape the heat in Rome and looking to quaff great quantities of Frascati wine.
Directly to the west is Fiumicino Airport (FCO).
The airport is sometimes called Ciampino–G. B. Pastine International Airport after a World War I airship flying ace.
COTRAL/Schiaffini operates buses from outside the terminal building to both Anagnina metro station and Ciampino Città railway station every 15 minutes. From the Ciampino train station, marked on the map, you can take a train to Rome Termini which takes 15-20 minutes. Going in the other direction, the train to Castel Gondolfo takes about 26 minutes.
Bus operators Terravision, Schiaffini and BusShuttle run a direct service to Roma Termini, travel time is about 40 minutes.
A taxi to Rome Termini should cost about 30 €.
You can get a private transfer from Ciampino Airport to your hotel in Rome for a reasonable prices as well.
Folks who've stayed there give La Villetta Suite a "superb" rating for a hotel near the airport. If you're not waiting for an early flight out you may consider staying in the Castelli Romani for a night or two.
If you are staying in Ciampino you might wonder if there's a hip place to sip and munch in the center of town. The newly opened "wine bar, cocktail Bar, Restaurant" called VapEat Experience might just do the trick. It's at Via IV Novembre, 7 in Ciampino.