There were actually several invasions by people from Essos
Invasion of the First Men - 12,000 years ago
At this time Westeros was settled by the Children of the Forest. But human settlers eventually arrived in Westeros from Essos through a land bridge that connected the two continents at the time called the Arm of Dorne. The humans were later given the name the First Men and arrived with bronze weaponry. First Men and Children of the Forest clashed for a long time, with the humans steadily advancing and cutting of the tress that gave the Children much of their power. The Children used their magic to shatter the Arm of Dorne, but it was too late.
The Children were eventually able to curb the invasion at Moat Cailin, laying down a magical calamity on the area called the Hammer of the Water, that shattered the area into bogs and swamps that separate North from South to this day. The result was a peace accord between the two races, and a friendship that lasted 4000 years that saw the First Men adapting the worship of the Old Gods and other customs.
Invasion of the Andals - 2,000-6,000 years ago
A second wave of human settlers arrived by sea who called themselves the Andals. They brought iron weaponry and the worship of the Seven. They would eventually invade six of the seven kingdoms, with only the Winter Kings being able to stop them at Moat Cailin.
Invasion of the Rhoynar - 1,000 years ago
Escaping the violent expansion of the Valyrian Freehold, the Rhoynar arrived in Dorne lead by their warrior queen Nymeria. There, Nymeria married Mors Martell, allying her people with his. The Martell-Rhoynar alliance would eventually conquer all of Dorne.
Invasion of the Valyrian Freehold - 500 years ago
House Targaryen of Valyria conquered the isle of Dragonstone and built a castle there, making it Valyria's westernmost outpost. A century later, Valyria itself is destroyed in an event called the Doom of Valyria, which was possibly a massive volcanic eruption.
Invasion of the Targaryens - 300 years ago
A few Valyrian houses headed by House Targaryen escaped the Doom by preemptively migrating to Dragonstone with one dragon, Balerion the Black Dread. There they stayed for 100 years, building up their forces and adding two more dragons to their forces, Vhagar and Meraxes.
From there, Aegon the Conqueror set out to invade the rest of Westeros, successfully uniting the seven kingdoms (with the notable exception of Dorne) under his Iron Throne. The rest is history.
Spoilers for "A Dance With Dragons"
The Invasion of Aegon VI and the Golden Company
Claiming to be Rhaegar Targaryen's son (presumed dead), Aegon VI arrives in Westeros backed by the Golden Company and conquers several cities in the Stormlands and the Stepstones.