If you haven't seen it, go do so now. Right now.
The Legend of Korra is the follow-up series. It's set many years after the adventures of the previous series, and most of those characters are deceased (Katara appears in a couple of episodes as an elderly woman), though their actions influence heavily this modern world. Time has moved on and now there are cars on the streets, electricity, radios and numerous other devices.
While Aang managed to settle the grievances between the various 'bending' nations ('benders' are people able to control one of numerous substances - earth, fire, water, air), there's now growing hostility between the benders and non-benders. Korra, the new Avatar (the only person who's able to control all four elements), arrives in Republic City initially to complete her air-bending training, but is soon involved in attempting to sooth relations between the citizens.
The main characters are older than they were in the previous series - around 16, I'd guess as opposed to 12ish in The Last Airbender - and there is a greater age range in the supporting cast too. Before there was only really Iroh, but now there's Aang's middle-aged son Tenzin, police chief Lin who's of a similar age and numerous others.
With the older characters comes a more adult tone. While The Last Airbender dealt with some weighty themes, it was generally a light and fun series. The Legend of Korra is about racism and terrorism. There are numerous electrocutions and a bad guy who will scare small children. Though there are some lighter moments ('you leave my dad's ex-girlfriend alone!') they are in the minority.
If you loved The Last Airbender, you'll love this. If you loved things like The Mysterious Cities of Gold, you'll love this. If you love well-written drama, you'll love this - assuming you can put aside any prejudices about it being a cartoon.
The Legend of Korra is currently available as a Region 1 DVD and region-free Blu-ray. Both contain all the episodes, though the Blu-ray has a much more extensive range of episode commentaries.