EVE is the substance in the BioShock universe which allows you to continue to use the plasmids you have equipped - similar to how mana or willpower works to magic in other games. It is blue in colour and can be found in hypodermic needles, called EVE hypos. EVE comes hand in hand with health in the world of Rapture; you can't survive effectively with low levels of either and as such, the EVE can be dispensed from said hypos via injection into the player's arm to deliver an EVE boost to replenish your EVE levels. As the growth of plasmids exploded in Rapture, splicers needed a means to keep their powers replenished, hence EVE.
EVE isn't always gained from EVE hypos. Users can also gain EVE snacks, cigarettes, coffee and with the help of gene tonics, EVE can also be obtained from alcohol, running water, health stations and numerous other sources, such as vending machines. EVE is usually readily available in surgical or medical areas, or in areas where the player must complete several plasmid related tasks.