From the FAA press release:

Under the final rule, the person actually flying a drone must be at least 16 years old and have a remote pilot certificate with a small UAS rating, or be directly supervised by someone with such a certificate. To qualify for a remote pilot certificate, an individual must either pass an initial aeronautical knowledge test at an FAA-approved knowledge testing center or have an existing non-student Part 61 pilot certificate. If qualifying under the latter provision, a pilot must have completed a flight review in the previous 24 months and must take a UAS online training course provided by the FAA. The TSA will conduct a security background check of all remote pilot applications prior to issuance of a certificate.

Here are some of the salient points from the summary:

• These rules apply to unmanned vehicles weighing up to 55 pounds.
• The drone must be flown within visual line of sight of the pilot (or an observer).
• Flying first person view (like with cameras on the drone streaming video into goggles worn by the pilot) doesn't count.
• Maximum allowed speed is 100mph.
• Can't fly more than 400 feet above ground.

Here's the specific rules for drone delivery from the summary:

Transportation of property for compensation or hire allowed provided that The aircraft, including its attached systems, payload and cargo weigh less than 55 pounds total; The flight is conducted within visual line of sight and not from a moving vehicle or aircraft; and The flight occurs wholly within the bounds of a State and does not involve transport between (1) Hawaii and another place in Hawaii through airspace outside Hawaii; (2) the District of Columbia and another place in the District of Columbia; or (3) a territory or possession of the United States and another place in the same territory or possession.

If a company or person wants to fly a drone in a way different from these rules, they can do so by applying to the FAA for a Certificate of Waiver, which if granted will give them a legal exception. The rules are a major step towards clarity in the vague world of drone law, though I'm certain there is much still to be decided and discovered in the full body of the rule.

Here, if you wish to delve through it, is the full rule.