[Sweet Basil, Genovese Basil, Mediterranean Basil, Ocimum basilicum (Mint family)]
This Basil is universally called for in European and American recipes. Formerly it was very easy to grow in season but lately it often falls victim to fusarium wilt, caused by a soilborne fungus, Fusarium oxysporum f. sp. basilicum.
Sweet Basil has decent keeping properties (almost a week if treated well). It is more aromatic and less sharp than Thai Purple Basil but the two can be used interchangeably in a pinch. I've even used Thai Purple Basil to make Italian Pesto because I can get piles of it cheap at ethnic markets while Italian Basil can cost several dollars an ounce in the supermarkets.
More on Basil.
Buying: Italian Basil is carried in just about every supermarket these days, but often at absurd prices. You may be able to procure it at a better price from farmer's markets when in season (Spring and early Summer). You can try growing it yourself (see next paragraph).
If the basil you bought If it's a little wilted, it can be refreshed by cutting the stem ends off and immersing the entire bunch completely in cold water for about 1/2 hour. Then dry in your salad spinner and wrap loosely in plastic.
Growing: Once highly reliable, growing basil has recently been a crap shoot. If their soil is infected with fusarium fungus they will grow to 8 to 12 inches high, then start dropping leaves and wilting. Fusarium infected soil should be sent to a landfill or sterilized by baking in the oven at high heat or by steaming because the fungus can hang on for another 8 to 12 years. Growing from seed is not a defense - the fungus is spread by infected seeds, and there is no sure-fire way to certify them fusarium free. There is also no effective treatment, but fusarium resistant varieties are starting to appear (Nufar Hybrids).
Storing: If fresh, Italian Basil can be kept loosely bagged in the refrigerator for almost a week. It can also be frozen for longer storage. Freeze in water to prevent drying out - basil is pretty much worthless dried, though preserving it in salt works a bit better.