Don't let the French spelling confuse you. The name of this building is pronounced "Fan-yul" or "Fannel". Getting it wrong is a good way to indicate you're not from around here, but it's also a great way of proving you're not French, which in my experience is a real boon.
Also, if you want to pass as a local, you should take a good look at the weathervane. It's a Boston landmark. Legend has it that in the war of 1812, suspected spies were asked what was on top of Faneuil Hall. Only a true local would be able to say it was a grasshopper.
The building was named after Peter Faneuil, the local merchant who paid to build it. It was meant as a place for local farmers to sell their goods, so their carts would stop crowding the streets, and also so you didn't have to walk near farmers.
However, it's the meeting hall on the second floor, which gives the building its nickname "The Cradle of Liberty". That's rather ostentatious - and dare I say, American - as a nickname, but it's not inaccurate. People met here to protest the Stamp Act. The first anti-tea tax meetings were also held here. Anti-British meetings occasionally drew crowds so large the building couldn't even hold them (at which point they all went to the Old South Meeting Hall).
During the siege of Boston, the meeting hall here was made over into a theatre. Since plays of any kind were banned in Boston under ordinary circumstances - something I think should still be a law, actually - this is yet another example of the British Loyalists sticking it to the rebels.