Most commonly, short for The Super Dimension Fortress Macross, a 36-episode anime series aired in Japan beginning in 1982 created by Studio Nue. A number of sequels and a theatrical movie followed. Macross was the first series to take the idea of transformable robots -- robots that reconfigured into vehicles -- seriously.
"Macross" is also the name of the giant starship which is the centerpiece of the show.
Macross is frequently confused with Robotech, an American series that stitched together and combined three anime series: Super Dimension Fortress Macross, Super Dimension Cavalry Southern Cross (Southern Cross), and Genesis Climber Mospeada (Mospeada). This writeup concentrates on the original Macross series alone.
In the year 1999 World War III is interrupted when a giant spaceship crashes on South Ataria Island. (Not Macross Island; that's the name given to the island in Robotech.) Duly shocked by the proof that there's at least one alien civilization out there, and it has big guns, much of Earth unifies under a one-world government.
Ten years later, a warlike race known as the Zentraedi (American Robotech spelling; sorry, purists!) tracked the battlefortress, now commandeered, retrofitted, and named SDF-1 Macross. The Zentraedi attempt to claim the fortress, and the bulk of the series follows the Macross' battles to survive as seen through the eyes of three main characters -- the Obligatory Anime Love Triangle:
Hikaru Ichijyo, a 16 year-old stunt pilot who enlists in the U.N. Spacy as a pilot of a Valkyrie transformable fighter.
Lynn Minmei, a 15 year-old aspiring singer and rather flighty primary love interest to Hikaru.
Misa Hayase, the reserved 19 year-old Chief Tactical Officer of the Macross.
Macross earned its American popularity thanks to Robotech; its storyline survived more or less intact.
A theatrical feature, Macross: Do You Remember Love, compressed and retold the story with high-quality animation. Other sequels include the OAV series Macross II (produced by a different company and considered to take place in a "parallel universe"), the OAV series Macross Plus, and the television series Macross 7.