What Fortune affects
Fortune increases drops from coal, diamond, emerald, redstone, lapis, nether quartz, nether gold and copper ore, from crops such as wheat, carrots, potatoes and beetroot, from melons, from gravel (flint), from leaves (saplings and apples) and from several other blocks. It does not affect iron or gold ore, which drop raw ore one-for-one regardless.
The three levels
Fortune I averages roughly 1.33× drops, Fortune II about 1.75×, and Fortune III about 2.2× — the exact count is a weighted random roll per block, not a fixed multiplier. On crops the benefit is a higher chance of extra seeds and produce; on ores it is more items per block.
Fortune on different tools
Fortune applies to a pickaxe (ores), an axe (apples/saplings from leaves), a shovel (flint from gravel and other drops) and a hoe (crops). A Fortune III pickaxe is the standard mining tool; a Fortune hoe speeds up farming significantly.