The Falcon is the main battle tank of the Eldar, capable of being equipped with a variety of weapons such as pulse lasers and shuriken cannons and doubling sometimes as a troop transport. Due to either the Eldar or Games Workshop being lazy, all Eldar tanks have the same body as the Falcon, just as the super-heavy Scorpion. (Because the Imperium has such a diverse and creative collection of chassis for their vehicles.)
Snarking aside, the Falcon occupies a similar slot in the Eldar army to the Razorback Transport as a tank that sacrifices transport capacity for more firepower in terms of a pulse laser and a heavy weapon of your choice, though this comes at a discount over the Wave Serpent. Falcons occupy the Heavy Support slot, which means it is competing with some of your other good options in 9th edition, like Dark Reapers and War Walkers. Given that it has a transport capacity of 6 though, it can quite competently house one squad of Aspect Warriors and even a Warlock or Farseer for suppot. Aside from the aforementioned competition in its slot, the Falcon faces two other problems: firstly, in an army of specialist units the Falcon is a more generalist option. With the pulse laser able to take out vehicles with its high strength and AP, it'll not completely cut through armour saves like a Bright Lance will, for example. In addition, it has relatively low volume of fire too, meaning any support in terms of psychic buffs will definitely help amp it, but it will have a relatively lower impact than other units, such as Fire Dragons or even a Night Spinner.
A big and obvious bright side is the anti-gravitic generators, which give the Falcon its amazing speed and agility are prime examples of the Eldar's technological mastery. Thanks to these powerful devices a Falcon can achieve speeds of up to 800 kph and soar to altitudes much higher than the equivalent Imperial craft, giving it limited flight capability. The Falcon's crew of pilot and gunner will themselves be veteran pilots, having learned their craft either on Eldar Jetbikes or along a similar Path, and take full advantage of these abilities.
Unfortunately, thanks to a certain Irish shitgremlin, a Falcon can just be as susceptible in being disabled with a simple rock in its engines. I guess it was a bad idea to design a tank with such an obvious weak spot eh?
From a lore perspective its name is a rough translation of Faolchu, who was the consort of the Great Hawk that retrieved Vaul's sword and gave it to one of the Eldar hero bigwigs during the War in Heaven.