The Savoy is a region on the northern border of the Alps; the historical borders cross the current French-Italian border, although the area currently known as the Savoy is entirely within the borders of France. It is -- very broadly -- defined by the borders of the feudal County of Savoy (1003-1416).
With the rise of the Holy Roman Empire, the territory of Sapaudia was given over to Humbert the White-Handed in 1003; while technically the County of Savoy was part of the Kingdom of Arles, they maintained a fairly independent existence, detaching from Arles in 1361. Over time the House of Savoy expanded their territory to rule the Kingdom of Sicily from 1713-1720, and when the War of the Quadruple Alliance forced them to trade Sicily for Sardinia, they continued to expand their territories from their new home base. In 1861 Victor Emmanuel II of Italy (formally Victor Emmanuel II of Savoy) was crowned King of Italy; the House of Savoy maintains a presence in Italy today, and the title of Duke of Savoy is still carried by Prince Amedeo.
The area known as the Savoy, however, did not expand to cover all of Italy; while the exact borders changed over time, it generally extended from Lake Geneva in the north to the Dauphiné in the south. While it originally included parts of western Italy and southwestern Switzerland, it is primarily contained in the modern French départements of Savoie and Haute-Savoie; at one time it extended so far as to include Nice within its borders, an expansion that gave it access to the Mediterranean Sea.
In 1848 the French Second Republic attempted to annex Savoy, taking the capital city of Chambéry briefly before being repulsed. Realizing that that they had something the French wanted, the Kingdom of Sardinia cut a deal with Napoleon III: the French would provide military support to Sardinia to help them in their fight with the Austrian Empire, and in exchange they would get the County of Nice and Duchy of Savoy, along with a trade deal.
This 'secret' deal quickly became widely known and was rather controversial, both locally and internationally. While it was formalized in the Treaty of Turin on March 24, 1860, it was done with the caveat that this would only happen if the populations of Nice and Savoy agreed to the deal. A vote was held in Nice on April 15-16 and in Savoy on April 22-23, and the result was a 99.8% vote in favor of joining France. There were claims of vote rigging, and the locals complained that the vote was limiting, being a simple yes/no vote on joining France, ignoring the options of independence or allying with Switzerland, but there was no revolt, so the Savoy was now part of France.
The event of World War I ended any special treatment the Savoy had under the Treaty of Turin -- it had been guaranteed military neutrality and a made a duty-free zone -- and while France's control of the Savoy was often a matter of contention, it was a matter of politics, not guns. There are still small, peaceful, separatist movements within the Savoy, and opinion polls suggest that large portions (~40-50%) of the Savoy population would support an independent Savoy. This is unlikely to happen in the foreseeable future.