So far, the other 9 answers are incorrect.
Flair is a 9 time(even though many say 10, the 10th was NOT a World title regin) NWA World champion. The two other title changes people refer to DID NOT TAKE PLACE. Flair simply allowed the person to leave the arena with the title, though he never lost the match. Flair is an offically 8 time champion, but he and Harley Race did a title change, where Flair was pinned, while doing an overseas tour. Since their was apin invovled, Flair is a 9 time NWA World Champion.
Flair is a 2 time WWF World Champion.
Flair is an 8 time WCW World Champion.
In all actualness, Flair has been a legit world champion 19 times. 1 of those was an unofficial NWA title change with Harley Race. Flair also has two matches where the man he faced just walked out with the title because he was afraid of riots but Flair did not lose the titles.
Also should be noted.... Midnight Rider did pin Ric Flair for the NWA World title. When he refused to disclose who he was the title was returned to Flair. If wrestling promotors had followed kayfabe better, they would have actually counted Midnight Rider as a world champion and their would be a note stating he was stripped of the title due to not following NWA Bylaws. However, Since wrestling is a work, they already knew they wasnt letting Dusty(Midnight Rider) keep the title so they simply gave it back to Flair. When the title was returned to Flair, it should have counted as a new title reign, which would give him another to count. Unfortunately, NWA didnt have that foresight.
So it boils down to Flair having 18 legit title reigns.
1 unofficial title change with Race that DID involve a planned pin.
2 times Flair let his opponent keep the title out of fear.
1 time as NWA International champiom(NOT a World title reign)
So, the answer is 19 planned title reigns, 18 with approval, 1 without!
If you want to get technical, he has also held several "World" Tag team titles as well, so the number could be higher, but I felt you were asking about his single title reigns.
25+ year fan, 10+ years IN the wrestling business.