The chain or boom is a fairly common means of defending harbors or waterways from antiquity to modern times. See, for example, the anti-submarine chain that was constructed at Fort Wadsworth, Staten Island, New York in the twentieth century or the much earlier example in Aeneas Tacitus, who claims a man betrayed the city of Chios in the 4th c. BC by convincing the local townspeople to bring in the chain of the harbor for maintenance. Made of both wood and iron, the chain stretched across a body of water between two defensive points.

The famous chain which you reference is the chain of the Golden Horn, recorded to have been used only four occasions—the sieges of 821, 969, 1203, and 1453—, although I suspect it saw much more activity then that. In an excellent article on harbor chains, Benjamin Kedar draws attention to the Islamic geographer Ibn Hawqal, who in the late 10th c. observed the chain being used to require that ships visit the custom house, presumably so the tax man could get his due. Joannes Phocas, an Byzantine pilgrim visiting Beirut in 1185, noted in his observations of the city the strength of the city's great harbor chain as a good sign of the town's defensibility. Contrary to what we would assume, the chain didn't necessarily just hang above the water, but often floated! Nor was the chain expected to stop a ship from moving into a harbor all on its own, but rather was meant to slow and delay ships so that other defensive measures could be taken, either by land or by sea. In 1453, for example, the Byzantines placed ships behind the chain to help ward off the Ottoman fleet.

There were ways around the chain. Ships could shift their weight so that the chain was pushed under the boat's bow, allowing the ship to then tip over to the other side or use its weight to try and snap the chain itself. In one of the sagas, Snorri records that Harald Hardrada escaped the harbor of Constantinople in this way—having his men rush from one side of the boat to the other. But, such heroic efforts took time and great risk that, unless you happen to be a hero in Snorri's saga, most were unwilling to take. More often ships would attempt to pressure the chain until it snapped or seize control of one of the towers from which it hung.

Much like a urban center's wall, the chain was a defensive instrument whose health was a point of pride to the community; thus it is unsurprising that maintence of the chain was often an importance focus of in harbor towns. In the fourteenth century, New Rochelle created a special office of warden for their tower chain and the notarial registers of Marseille, rich in detail that they are, reveal that the guardianship of the chain became a focal point of dispute rather regularly. Indeed, the chain of a city was often regarded as a trophy that conquerors took and brought back to remember their great victories. After breaking Pisa's harbor open in 1287, the Genoese placed the "irons of the chain" inside the cathedral church of San Lorenzo. After all, the remnants of a chain are much easier to display alongside a banner than, say, the broken fragments of stone that might come from a wall.

If you are interested in reading further, I'd recommend Benjamin Kedar, "Prolegomena to a World History of Harbour and River Chains," in Shipping, Trade, and Crusade in the Medieval Mediterranean (2012), 3-38.

Just an aside of the great responses provided, but didn't Ruyter 1667 raid on the Thames burst througj chains raised to defend the river ? Meaning such a practice of defense wasn't discontinued until waaay after 1543?