the first 7 seasons Mighty Morphin Power Rangers through Power Rangers in Space, told one continuing story. The first 3 seasons they were the Mighty Morphin Power Rangers, about half way through the second season half the team left and were replaced with 3 new rangers who assumed their powers. Season 3 they got new powers but kept the same costumes (and could also become "ninjas") During season 3 the original pink ranger was left and replaced. Towards teh end of that season the big bad reversed time and made all the rangers kids, so they didn't have powers. Zordon (their mentor) asked a group of "Alien Rangers" from the aquatic planet Aquitar to come and help stop Lord Zedd and Rita Repulsa. This was the first time there was an entirely "new team" of rangers, this paved the way for the yearly rebranding/new powers. They reversed teh time reversal but in the process totally lost their powers, so they got a new source of powers, and a new yellow ranger (the previous one stayed in africa to help animals). half way through this season (Power Rangers Zeo) the original Red Ranger came back and became for a brief time the Gold Ranger (who was actually a black costumed ranger) the original blue ranger (who at the time was just the "tech advisor") left the show (because a lot of the crew were super homophobic). The following season (Power Rangers Turbo) they got new powers (with out any real explanation). Half way through this season the existing Rangers (minus the current blue ranger (who was a child)) left the show and were replaced by new characters. They in turn went off into space and got new powers after their powers were destroyed. During htat season (Power Rangers In Space) they wrapped up the over arcing storyline and killed Zordon and destroyed pretty much every villain the show had seen, also turning the more "human" villains into good people.
The following season (Power Rangers Lost Galaxy) was somewhat tied into the previous one, as the characters came from the same city, and they used Alpha (the robot) and the megaship (space craft) from the previous season). Half way through that season there was the first ever "death of a ranger" as the pink ranger "died" allowing the previous seasons's villain to become the pink ranger.
Every season after that tended to be its own city/storyline though most seasons did have a team up.
the 500th episode of Power Rangers happened in the season Power Rangers Dino Thunder and it was in that episode that they showed that every season was in fact in the same universe.
Rita Repulsa (the original villain) became a force of good and actual created the powers for the Power Rangers Mystic Force.
The only season thats NOT in the same continuity/universe is Power Rangers RPM as they said it was an alternate reality.
The current season Power Rangers Megaforce has a lot of call backs to the original series, including having their mentor be a student of Zordon.
The next season Power Rangers Supermegaforce will have lots of call backs to the entire series, including allowing the current team to use former rangers powers and weapons. There will be a big anniversary episode where 11 previous rangers come back for a big huge battle, there are also talks of other episodes with other characters from the series.