Isolated and heavily fortified, Cobra Island is the prime headquarters of the evil Cobra organization. In some adaptations, this island is shaped like the Cobra logo.
Marvel Comics continuity
Cobra's base was created by the machinations of Professor Appel, a Crimson Guardsman. Appell tricked the Joes into triggering a fault line beneath the Gulf of Mexico, resulting in a massive upthrust of the seabed. Despite a valiant effort, the Joes were unable to remove Cobra forces from the newly-formed island. Cobra lawyers had secured the organization's recognition as a sovereign state from various governments, which resulted in the immediate withdrawal of the Joe forces from the island.
On the island, there was a Terror Drome fortress, an airfield, surgical hospital, and a luxury hotel for dignataries (who were for the most part third-world dictators looking to purchase weapons). It was a heavily fortified and armed island, with bunkers, anti-aircraft artillery, missile batteries, armored vehicles, combat aircraft, and Cobra soldiers. A "land-locked freighter" on the island was a storehouse of advanced Cobra technology (e.g. the Cobra space shuttle launch facility and the orbiter). Upon their retreat from Springfield, this 5 sq. mile island underwent a significant increase in its population due to the civilians that were evacuated from Springfield. At the center of the island was an active volcano, created by the technology used to raise the island. It also has a swamp with trained crocodiles and a rapidly grown tropical jungle.
Eventually a civil war erupted between Serpentor and Crimson Guardsman Fred VII (who masqueraded as Cobra Commander). The war ended up pulling both G.I. Joe and the Iron Grenadiers into the conflict. The Joes were ordered by Washington DC to assist the island's constitutional monarchy (Serpentor), while Destro kept his Iron Grenadiers neutral. Serpentor's faction was about to win the war, when he was assassinated by Zartan. Destro was allowed to leave the island with the Baroness, which was what he really came for. The real Cobra Commander later returned and regained control of Cobra, imprisoning and then killing most of those who opposed him. After the Trucial Abysmia War with the Joes, Cobra Commander took his organization global and largely abandoned the island. Later on the last vestiges of Cobra deserted the island.
Devil's Due Comics continuity
After the defeat of Cobra in 1994, the United Nations annexed the island. A revived Serpentor made the island the base of operations for the Coil, his personal army. The island by then had a greatly-changed geography, with a fortified harbor surrounded by towering cliffs. After Serpentor's defeat, the United States sold the island to the "Gen-Tek" research firm, which the Joes discovered to be a Cobra front. One run by the Cobra businessmen, Xamot and Tomax. For a time it served as Cobra's headquarters, after which they abandoned the island and moved to the Monolith Base. When the Joes destroyed the island with a nuclear weapon, the remnants of the Coil were on the island.
In the Sunbow animated series, Cobra Island appears several times, but without much description as to its location, and with differing appearances. Its primary appearance matches that of the second season's opening credits: a swampy island covered with jungle terrain and several clear and low-lying areas. At the center of the island is the massive Cobra Terror Drome, which is situated atop a rocky cliff and converted into an inner sanctum for troops and vehicles that overlooks a long beach.
In G.I. Joe: The Movie, the island is destroyed when the Terror Drome is infiltrated and destroyed by Sgt. Slaughter and his Renegades. Serpentor and the Cobra High Command survive in an underground bunker, which is all that survives the explosion. Cobra Island is presumably abandoned by Cobra following this event.