Q: What is a Special Jounin?
A: I'm not quite sure, but I do know that is it an actual rank, below that of Jounin and above the rank of Chuunin. One theory is that a Special Jounin isn't quite good enough to be a full Jounin, but excels in a single field. Although it has never been stated, there is nothing to suggest that Special Jounin take on the role of the sensei of a Genin team.

Q: Do other Hidden Villages have Special Jounin, or is this a Konoha thing?
A: One of the ninja from Iwagakure (Hidden Village of Stone) in Kakashi Gaiden (volume 27 of the manga, not animated yet) was a special jounin. That's the only non-Kokoha special jounin I know of.

Now, on to the profiles. Hayate gets to go first because I love him. ^_^ The rest go in order of appearance.
This page is a work in progress. Particularly the episode numbers need work. It takes low priority.

A note about ages: Unless otherwise noted, the numbers are from the first character databook. The second databook adds a year to the age of the characters that appear in both books.

Name: Gekkou Hayate / 月光ハヤテ
Age: 23
Blood type: AB (Rh-)
Height: 175cm
Weight: 58kg
Genin at age 12.
Chuunin at age 13.

Mission History:
96 A-rank missions
407 B-rank missions
83 C-rank missions
66 D-rank missions

Special move: Mikazuki no Mai / Dance of the Crescent Moon [三日月の舞] (A-rank ninjutsu)
Appears in episodes 37-54

Name: Ebisu / エビス
Age: 28
Blood type: A
Height: 181,4cm
Weight: 67,5kg
Genin at age 10.
Chuunin at age 17.

Mission History:
1 A-rank mission
115 B-rank missions
284 C-rank missions
96 D-rank missions

For a Special Jounin, Ebisu is kind of weak. However, his skills in Ninjutsu, Taijutsu, Genjutsu and chakra control are near Jounin level. He is very intelligent, but a bit lacking in speed, strength and stamina.
Appears in episodes 2, ((adding the rest when I get there)), 80
Name: Morino Ibiki / 森乃イビキ
Age: 27
Blood type: A
Height: 193,5cm
Weight: 88kg
Genin at age 10.
Chuunin at age 15.

Mission History:
3 S-rank missions
79 A-rank missions
301 B-rank missions
297 C-rank missions
181 D-rank missions

Appears in episodes 23-((to be updated)), 68, 80

Name: Mitarashi Anko / みたらしアンコ
Age: 24
Blood type: A
Height: 167cm
Weight: 45,8kg
Genin at age 10.
Chuunin at age 12.

Mission History:
2 S-rank missions
89 A-rank missions
398 B-rank missions
86 C-rank missions
72 D-rank missions

Before Orochimaru left Konoha, he was Anko's sensei. He taught her a few forbidden techniques..

Known techniques:
Sen'eijashu / Hidden Shadow Snake Hand [潜影蛇手] (C-rank Ninjutsu)
Soujasousai no jutsu / Double Snake Suicide [双蛇相殺の術] (A-rank ninjutsu/kinjutsu)
Appears in episodes 24-((something)) 55-56

Name: Shiranui Genma / 不知火ゲンマ
Age: 29
Blood type: A
Height: 181cm
Weight: 71kg
Genin at age 10.
Chuunin at age 13.

Mission History: (second databook)
2 S-rank missions
74 A-rank missions
454 B-rank missions
220 C-rank missions
165 D-rank missions

Judge of the Chuunin Selection Exam finals.
Appears in episodes 55, 58-68, 80, 110-111
Name: Namiashi Raidou / 並足ライドウ
Age: 31
Blood type: AB
Height: 183cm
Weight: 75kg

Bodyguard to the Third Hokage during the Chuunin Selection Exam finals.
Appears in episodes 55, 60, 63, 65-68, 110-111
Name: Yamashiro Aoba / 山城アオバ
Age: 30
Blood type: O
Height: 178cm
Weight: 66kg

Aoba takes part in the battle at the end of the Chuunin Selection Exam finals, showing off some fire jutsu. He appears again in episode 83, and is finally named in episode 98. He has never been listed in the end credits.
Appears in episodes 55, 67, 79-80, 83 and 98.
Name: Inuzuka Tsume / 犬塚ツメ
Age: 36 (Second databook)
Blood type: B
Height: 163cm
Weight: 51.5kg

Kiba's mother. Wild and impatient. Her favorite food is fillet steak.
Appears in episodes 79 and 121.
Name: Toriichi Kumade / 酉市クマデ
Age: 29 at time of his death

Toriichi Kumade was on a spying mission to Kirigakure when he was killed by Zabuza, who at the time was part of Kirigakure's ANBU.
Appears in episode 7 (flashback scene)