Delta's Frequent Flyer scheme is called Skymiles: it is a pretty normal scheme, with points earned by a combination of fare paid plus the distance, with a class of travel multiplier.
A major perk of the programme is that miles do not expire: indeed, it is the only major US airline without mileage expiration.
Any member with even the slightest status of Silver and above gets a 50% bonus on miles in anything other than cheap economy seats.
Delta Airlines Domestic First Class on an Airbus A320
You may also come across Delta's SkyBonus scheme – which is a 'sort of' addition to the skymiles frequent flyer scheme. It is in effect a free Frequent Flyer scheme designed for small to mid-sized companies that aren't quite large enough for their own account with Delta.
Occasionally you may hear about the Million Miler scheme. It doesn't offer a lot, but if you rack up one million Medallion Qualification Miles over a lifetime you are awarded complimentary annual Silver Medallion status, two million gets you Gold, and so on.
Silver level also has Elite Skyteam status: which doesn't mean a lot.
Delta regional jet at Minneapolis
The main perk of this level is the abaility to book a free upgrade to Economy Comfort on Transcontinental Routes.
Gold level also has Elite Plus Skyteam status: which is really worth it: it gives you Skyteam lounge access across the network.
This comes with all the Gold Medallion perks, plus on reaching this level you get a choice of 4 Systemwide Upgrades, 20,000 miles, or 4 Delta Sky Club One-Day Passes. There is also the nice perk of getting International Economy Comfort Seats for free.