How many groups of Power Rangers have existed?

It's difficult to define "groups" as a couple of the teams overlap a lot, or some actors leave the show and the role is re-casted. For Example, The Turbo Rangers are most of the Zeo Rangers, whom are most of the Mighty Morphing Power Rangers in the first place.

Team 1 through 5 are slight cast, costume and setting changes, considered the Zordon Era. So it can be considered 7 seasons of a single show. (Alien Rangers being a filler mini-season). Everything forward of Lost Galaxy were pretty stand alone seasons/teams, until Samurai and Super Samurai, which are two seasons of the same team. The same for Megaforce & Super Megaforce, and Dino Charge & Dino Super Charge.

Given that, there are 20 distinct major groups (From Wikia):

1. Mighty Morphin Power Rangers
2. Aquitar Rangers (Might Morphin Alien Rangers)
3. Zeo Rangers
4. Turbo Rangers
5. Space Rangers
6. Galaxy Rangers
7. Lightspeed Rangers
8. Time Force Rangers
9. Wild Force Rangers
10. Ninja Rangers (3 intertwined teams: Wind Rangers, Thunder Rangers, & Green Samurai Ranger)
11. Dino Rangers
12. SPD Rangers
13. Mystic Rangers
14. Overdrive Rangers
15. Jungle Fury Rangers
16. Ranger Operators
17. Samurai Rangers
18. Mega Rangers
19. Dino Charge Rangers
20. [TBD] Ninja Steel Rangers, probably will have a unique name to not be confused with the Ninja Rangers

Spread among 23 individually titled series:

1. Mighty Morphin Power Rangers (3 Seasons)
2. Mighty Morphin Alien Rangers
3. Power Rangers Zeo
4. Power Rangers Turbo
5. Power Rangers in Space
6. Power Rangers Lost Galaxy
7. Power Rangers Lightspeed Rescue
8. Power Rangers Time Force
9. Power Rangers Wild Force
10. Power Rangers Ninja Storm
11. Power Rangers Dino Thunder
12. Power Rangers S.P.D.
13. Power Rangers Mystic Force
14. Power Rangers Operation Overdrive
15. Power Rangers Jungle Fury
16. Power Rangers RPM
17. Power Rangers Samurai (Season 1)
18. Power Rangers Super Samurai (Season 2)
19. Power Rangers Megaforce (Season 1)
20. Power Rangers Super Megaforce (Season 2)
21. Power Rangers Dino Charge (Season 1)
22. Power Rangers Dino Super Charge (Season 2)
23. Power Rangers Ninja Steel [TBD]

*This does not include any secondary groups, evil clone groups, history/past groups (Like the Legendary Rangers (actors from past shows), or alternative power sets (like the Ninja Rangers for the MMPR, or Shogun Rangers for the Samurai Rangers).

This is also limited to the Americanized show, not the original Japanese shows that the scenes are cut from. Which is much longer, and disjointed, as it's a genre of multiple unconnected shows, not just a franchise.

Oh, and the Power Rangers Movie is set in an alternative timeline. The Series has a slightly different view of events during season 3. So make that 21 groups, but outside of continuity.

So, The Power Rangers are actually an American adaptation of a Japanese show called super sentai. They started releasing the English versions for american audiences in 1993 (MMPR) which was our first. In Japan, the equivalent was known as Ky?ry? Sentai Zyuranger, which was the 16th "Season" and group. Every time the original group switched suits was technically a new group in Japan. So MMPR, Zeo, etc are entirely differnt people in the Japanese series, much like it is now in america. As of now, I believe they are on their 40th series.