Sarutobi Asuma was a jounin from the Hidden Leaf Village and the sensei of Ino, Shikamaru and Chouji. At the age of nine he graduated from the Ninja Academy, and at the age of 12 he passed the chuunin exam. Apparently Asuma was also one of the Fire Country's Twelve Elite Guardians, shinobi specially trained to protect the Fire Country Daimyou. Asuma continued to wear the group's loincloth marked with the "Fire" character on top of it. Eventually he left Konoha behind to seek out his own path but returned at some point before the series' start.
Asuma, being the sensei of Team 10, let his genin enter the chuunin exams at the same time as Naruto and his teammates did. Asuma was impressed with his team's strength, and allowed them to participate. In the end only one of his genin made it into the final round, and actually became a chuunin, namely Nara Shikamaru. During the invasion and attack on Konoha, Asuma fought off and defeated the Hidden Sound shinobi that were following Shikamaru and Sasuke. Asuma fights mainly using knuckle blades, and uses his chakra to extend the blades to do damage outside of the physical blades' range. The extensions of the blades are invisible, so it may appear that he missed but seconds after the blade appears to have missed, a cut or wound will appear.
We also saw Asuma fight alongside Kurenai and Kakashi when Uchiha Itachi and Hoshigaki Kisame came to Konoha looking for Naruto. Asuma took on Kisame, but their battle came to an end and wasn't settled. Asuma was able to cut Hoshigaki Kisame, but Kisame however revealed some tricks of his own and was able to shred Asuma's arm with his sword Samehada. When Itachi moved to kill Kurenai, Asuma quickly jumped in to help. Itachi and Kisame fled after Itachi used his Tsukoyomi on Kakashi, and Maito Gai came to the rescue.
Sarutobi Asuma was born into the Sarutobi clan, and is the Third Hokage's son. He apparently had some problems with his father, but came to understand his place in time. Asuma always seems very relaxed, and is almost always seen smoking. He also had a relationship with Kurenai, whom he has a child with later in the series.
In Shippuuden, Asuma gets sent on a mission together with Shikamaru, Izumo and Kotetsu as one team of the Nijuu Shoutai. His team, along with 19 other four-man teams, was ordered to search for and, if possible, capture the two Akatsuki members who had entered Konoha. Eventually, Asuma's team tracked down the Akatsuki members, Hidan and Kakuzu, at a Bounty Station. After a fairly long fight there, Asuma loses to Hidan, who is immortal, and receives a fatal blow. Before dying, Asuma smoked one last cigarette and said some final words to his students. Later, Asuma appeared before Shikamaru, congratulating him for defeating Hidan and entrusting him the Will of Fire. He also told him to take care of his and Kurenai's unborn child, before disappearing in a puff of cigarette smoke.