It's now a city park called the At Meydanı (Horse Grounds) because of its function in Ottoman times.
When I first visited it in 1968, it was ringed with Volkswagen minibuses driven by Hippies on their way to Kathmandu. Today it is ringed by giant motorcoaches ferrying hordes of tourists of all ages to the principal sights of Istanbul. In high season, the buses form a solid wall around the Hippodrome, which is an unsightly pity.
The Basilica Cistern, (Yerebatan Sarnıçı) is beneath the little park at the northern end of the Hippodrome. Above the hidden cistern is a stone tower that was once part of the city's system of aqueducts.
Beside the stone tower is the Milion, all that remains of a triumphal gate that served as the zero-mile-marker on the road called the Mese (now Divan Yolu [map]), the Roman road between Constantinople and Rome.
Ayasofya (Hagia Sophia) is across the street from the stone tower, Topkapı Palace is just beyond Ayasofya, and the Istanbul Archeological Museums are next to Topkapı, down the hill bordering Gülhane Park.
During a visit in 1901, Kaiser Wilhelm II of Germany erected an elaborate temple-like fountain near the northeastern end of the Hippodrome as a gift to the sultan and his people.
Monuments decorating the Hippodrome include the 3500-year-old Egyptian granite Obelisk of Theodosius, brought to Constantinople by Emperor Theodosius in 390 AD.
You'll also see the spiral bronze base of a three-headed serpent sculpture brought from Delphi in Greece (the serpents' heads are in the Archeological Museum just down the hill).
At the southwestern end of the Hippodrome is the bare stone Column of Constantine Porphyrogenetus, dating from the 10th century.
Just west of the Hippodrome is the Binbirdirek Cistern("Cistern of 1001 Columns"), worth a look.