What does "Extra Virgin" on the label really mean?

It means that you ought know what's in the bottle. This means, of course, everything you need to know ought to be on the label. If the producer and seller don't show you the official olive oil analysis, name the olive or olives, tell you from which groves the olives came, how they were pressed, and where they were pressed, you should be thinking carefully about your purchase, especially without tasting it first.
100% Nocellara del Belice and
100% Nocellara dell' Etna
Extra Virgin Olive Oils

Cavallo Sicilian Wines

Fava Beans, Tangerine Marmalade, Caponata, and Dried Tomatoes in oil
Olive Patès:
Capers, Almonds, and Tomato

Cavallo Sicilian Wines Food Products

To be classified as "extra virgin," the oil analysis must show that the oil in the bottle or can has less than 0.8% acid. Because of costs, most producers buy very limited amounts of truly extra virgin olive oil with acid at 0.2% or less. Too often extra virgin olive oils are blends of lower quality oils mixed with higher quality oils to meet or get below the 0.8% maximum acid guideline to put "Extra Virgin" on the label, and then they are filtered to clear out bitter impurities. Most Americans know little about this, so they often unknowingly buy lower-quality, mass-produced olive oil made from unlisted olives from unlisted regions but labeled "Extra Virgin."
In sum, if you can't drink the "extra virgin" olive oil, you might want to reconsider your purchase or restaurant.