With the world's best and brightest scientists together at the bottom of the sea with no constraints, it was only a matter of time until the population made a number of breakthroughs. The defining breakthrough was ADAM. Generated by a deep sea parasite when attached to a host, the parasite produced new stem cells healing the host. The new stem cells allowed the host to take on external genetic modifications that take over their normal cell structure (like cancer). These genetic modifications ranged from medical cures to pure vanity changes. The genetic modifications of Rapture come in two flavors, Plasmids and Gene Tonics.
ADAM is a second currency in this place and is required to "keep the cogs of Rapture turning", to quote Andrew Ryan. As the citizens of this world became more dependent on it, they turned on one another, attacking their family members, friends and co-workers in the hopes of obtaining more.
Once enough ADAM is ingested, the person in question becomes known as a Splicer -- highly delusional, aggressive, and violent.
 Obtaining ADAM and Harvesting vs Rescuing Little Sisters
Little Sisters hold the key to ADAM. With their extraction tool, they take ADAM from deceased Splicers, according to the will of their 'creators'. From the perspective of a Splicer, ADAM is obtained by killing a Little Sister. However, it is not so simple, as they are protected by Big Daddies. As such, Splicers never succeed in acquiring the substance in this way. For you, however, this is feasible.
Upon killing the Sister's Daddy, you, the player may harvest her (kill her, or 'free' her as one character in the game phrases it) for 160 ADAM, or rescue her (return her to a normal state) for 80 ADAM -- the choice being up to you. While it would seem obvious for some, every time you rescue three Little Sisters you will receive a reward from Tenenbaum -- 200 ADAM and a variety of other items (typically costly ammunition, Plasmids, First Aid Kits and/or EVE). While in the short-term harvesting is more beneficial, in the long-term more ADAM is obtained (480 versus 440).
Note that there is also an Achievement for Harvesting or Rescuing all Little Sisters, so once you choose how to deal with them you should be consistent throughout the game.