A-Rank is one of the six classifications. It has several purposes and is preceded by B-Rank and followed by S-Class. One of the purposes may be an alternate ninja rank. A-Rank is classified for Jonin who are the most skilled shinobi in the village not counting the Kage. This is not very common however.
The more common use of "A-Rank" are missions. These missions are carried out by average Jonin and usually involve asassination. One mission was to escort Tazuna, the bridge builder. Team 7 carried out this mission as their first "real" mission. This mission was supposed to be a C-Rank but however it was later elevated to be an A-Rank mission because of the risks of Gato and his missing-nin, the Demon Brothers, Haku, and Zabuza. Other missions involve in Part II, to find Sasuke, assigned by Team Yamato.
Another purpose for "A-Rank" is techniques. These are jutsu that are quite useful in battle and in other. They are Kage or Jonin level. Some are even Kinjutsu.