Olive oil has been used since antiquity as a recipe for health, a base for cosmetics, nutrition and fuel.
Olive oil is obtained by crushing the whole fruit of the olive tree and then extracting the oil with a centrifuge, yielding 12-22% oil, depending on the condition of the tree, weather and growth conditions. The yield and quality are generally best in a dry year--too much rain waters down the flavour and reduces the oil content of the fruit, as well as letting pests breed. Olives destined for oil should be picked just before full ripeness to get the perfect balance between oil yield, higher when ripe, and acidity, lower when unripe. Mechanical picking bruises the fruit and reduces the quality of the oil. The olives should be processed immediately after picking and not exposed to moisture while they wait to be pressed. Traditional methods mandate that no heat be applied to facilitate the separation of the oil from the water and solids so the indication "cold-pressed" is a good sign. The solid waste of the pressing becomes olive cake.
The oil comes in various shades ranging from golden yellow to dark green. As a rule of thumb, the darker the oil, the higher the quality. Freshly pressed oil will usually be rich, dark green, a bit cloudy and taste stronger. Yellow oil tends to be of lesser quality but more suitable for frying. The story about olive oil being unsuitable for frying is mostly a myth. Lesser quality oil (sold as "light") is ideal for frying because of its discreet taste, low absorption and high smoke point. Extra virgin enhances the flavour of sauteed food but should not be used for deep frying. It's perfect for foods such as fried eggs, onions or mushrooms. Uncooked it's pretty much the only oil worth using in a salad.
The olive oil you see in the shops will almost always be extra virgin, meaning it has acidity of less than one degree, defined as containing less than 1% oleic acid. Other varieties are virgin, with up to 2 degrees acidity (1.5 by some definitions) and classic or pure (2-3 degrees). The former two are from the first pressing whereas anything else is the result of additional processing. Anything more acidic than that is inedible and most oil-producing countries will not allow it to be sold without refining to reduce the acidity. If you ask a Mediterranean cook, anything not extra virgin is barely fit for engine grease. Now, this is not saying that 0.1 and 0.9 acidity are the same... they're not. An olive farmer prides himself on producing oil with 0.4 or less acidity. If it's 0.2 or less, he has bragging rights for the best oil in town.
Oil should not be stored in plastic containers or bought in such since it can absorb chemicals in the packaging, especially PVC, and you have no idea how long it's been in contact with the plastic. Lacking the glazed clay jars of old, glass and stainless steel are best. Outside the Mediterranean basic oil is mostly sold in glass bottles containing one liter or less. Where oil is used a lot, it's more common to use metal containers of 4 and 17 liters.
The origin of the oil does matter. Being most familiar with Greek oil, I can vouch for oil from Laconia, Lesbos and eastern Achaia. Olives, like wine, produce different grades of oil and fruit depending on the local micro-climate. Generally speaking, though, I find Greek and Turkish oil to be of higher quality than most Italian and Spanish oils on the US market, though it is my understanding that parts of southern Italy and Andalusia produce some very fine oil. Most exported oils come from the two largest producers, which are Italy and Spain. Italian or Spanish export oils are usually a blend in which oil from Turkey, Greece, Tunisia, or other countries is added in order to improve the quality.
Olive oil keeps very well if stored at temperatures under 15°C/59°F in a well-sealed container and away from light. Under these conditions it should be used within six months but can last two years if kept refrigerated. It has a higher freezing point than any other oil. Pure olive oil congeals at 2°C/36°F because of its high stearin content; that's also a very good indicator when checking for adulteration...other oils will still be liquid and animal fats will already have congealed. After some time in storage, some of the residual vegetable water may settled to the bottom. Straining the oil through cotton wool can help.
Cost: Good olive oil is known for being expensive. This is because the traditional methods required to produce high quality oil are labour-intensive. A good picker will gather no more than 100kg/220lbs of olives in a day and, with a typical oil yield of 15%, that means 15kg (unlike other liquids, olive oil is measured by weight in the Mediterranean). Add the cost of pressing the oil by the less productive method of cold-pressing and there really isn't that much that comes out of a day's work. Spending most available daylight hours up a tree in the middle of winter (olives are harvested in December and January) is something that deserves its pay. So is spending the rest of the day kneeling under said tree picking out impurities like twigs and bugs (millipedes love olive trees and taste nasty--you don't want them in your oil). The price is not unreasonable.