I have eaten paella in Spain; I then ate it in Italy, but it didn't taste like the one in Spain.
What is the ingredient that makes the Spanish paella so particular? I remember they used a particular sauce for the meat (I have eaten both the Valencian paella and the seafood paella); if that is what makes the difference, what are the ingredients for that sauce?
The most prominent flavors in authentic paella are pimenton (smoked paprika), saffron, a sofrito of onion and tomato, garlic, and the broth.
When I studied abroad in Spain, I lived for 3 months with an elderly Spanish woman, who originally lived in Cuba (so I had some amazing food). Along with teaching me how to make tortilla (Spanish tortilla), she said that saffron was the special ingredient that made the paella special. I should add that to get the best flavor, you need to get real saffron, not the artificial kind. The only problem is real saffron is pretty expensive.
The secret ingredients in paella are the socarrat and cooking with wood. Socarrat comes from the part of the rice browning and burning at the bottom of the pan, and thus slightly smoking the rest of it. Another part of the subtle smoked flavor comes from the wood.
The special ingredient of paella is saffron (as everybody else already confirms).
The best way of preparing saffron for your paella is to soak it in warm water at least 30 min. before you add it to any dish. Take care not to mix it with your broth as you are not going to use all of it. When the rice is ready to be cooked, first add one measure of boiling broth, then add the saffron infusion, then add the rest of the broth you need to measure out.
When buying saffron, look for threads, those are the stamps of the flowers. Don't buy the saffron powder. The powder is cheaper, but you can't be sure it wasn't adulterated. Saffron is very expensive, as it's very labor intensive.
Really real paella is cooked on vines. That will give a smoky flavor. But I don't think you'll find that in a restaurant.
