Sequel Series

An entire series that follows on chronologically from the predecessor. Marked by a change in cast within the universe, while maintaining the same general rules.

Examples of Sequel Series include:

Anime and Manga

Eureka Seven is followed by Eureka Seven AO.
Similar to the Star Trek franchise, the Gundam franchise is made of this trope. Just looking at the Universal Century and excluding OVAs, you can come up with four different ~50 episode series: Mobile Suit Gundam, Zeta Gundam, Mobile Suit Gundam ZZ, and Victory Gundam. When one adds in the OVAs and movies, the UC era alone has well over a dozen entries in it.
Where most Digimon series are Alternate Continuities, Digimon Adventure is directly followed by Digimon Adventure 02.
And Digimon Xros Wars gets one, too!
Naruto Shippudden takes place 2–3 years after the original Naruto anime.
Umineko no Naku Koro ni, and boy did it get worse.
Dragonball Z, which is almost twice as long as the original.
Leo The Lion covers the last third of the original manga of Kimba the White Lion after the 60's anime ended.

Literature

Dean Koontz's Frankenstein. Set in modern New Orleans and follows the monster and Victor Frankenstein after a 170-year Time Skip.
Rick Riordan's Percy Jackson and The Olympians is being followed up by The Heroes of Olympus.
Warrior Cats spawned THREE six-book series after the original six books, not to mention the graphic novels, Special Editions, and other "extras".
Leven Thumps is followed by a trilogy called Beyond Foo.

Live-Action TV

Star Trek: The Next Generation, Star Trek: Deep Space Nine, and Star Trek: Voyager were all concurrently running sequel series to Star Trek: The Original Series. The prequel was Star Trek: Enterprise.
Similarly, Stargate SG 1, Stargate Atlantis, and Stargate Universe are all sequels to Stargate.
90210, the sequel to Beverly Hills, 90210.
- Likewise, Melrose Place has a new incarnation. Both revivals are arguably In Name Only, as they fit easily into the typical 'modern drama' template as The OC or Gossip Girl.
Power Rangers moved from Oddly Named Sequels to Mighty Morphin Power Rangers (Mighty Morphin' Alien Rangers, Power Rangers Zeo, Turbo and In Space) to this in its seventh season, Lost Galaxy.
Rick Steves' Europe, to Travels in Europe.
Degrassi Junior High was immediately followed by Degrassi High, and then, ten years later, by Degrassi the Next Generation.
In 1988, the original |Mission Impossible was revived with Jim Phelps leading a team of new agents—including the son of one of the original team members.
Technically speaking, all four series of Blackadder were separate shows. The 1983 original The Black Adder was followed by Blackadder II in 1986, then Blackadder the Third in 1987, then Blackadder Goes Forth in 1989.
Garo is followed by Garo: Makai Senki.
Devoted fans of the original Battlestar Galactica will deny it ever existed, but it got a brief resurrection after its cancellation as Galactica 1980.

Video Games

The Mega Man series has 6 sequel series, including Mega Man X, Mega Man Zero and Mega Man Star Force. See Video Game Long Runners.
Apollo Justice: Ace Attorney, the sequel of the Phoenix Wright: Ace Attorney trilogy. Meanwhile, Ace Attorney Investigations: Miles Edgeworth is a Midquel series, being set between Phoenix and Apollo's sagas.
Ace Combat is a prequel series. The first two games didn't have much to do with each other or the following games, so it was Ace Combat 3 Electrosphere that laid the foundations of Strangereal. The problem was that it was set in 2040 and the developers wanted to return to the Present Day setting of the first games. So, every game since then has been a build-up to Electrosphere, at least until the Continuity Reboot with Ace Combat Joint Assault and Ace Combat: Assault Horizon (particularly vitriolic fandom voices claim this happened because the timeline was getting dangerously close to Electrosphere and the developers have no idea where to go from there). They even managed to make a prequel to the prequel series with Ace Combat Zero, set in 1995.
- Of course, Electrosphere just had to be macekred beyond any recognition for the US release, and it never reached Europe at all.
The Metroid Prime saga is an Interquel series, taking place between the first and second 2D games.
Modern Warfare, to Call of Duty.

Web Original

BikdipOnABus has his LP of a capless, cannnonless, coinless playthrough of Super Mario 64 and later, a capless, cannonless, all-coins playthrough of it.

Western Animation

Beast Wars was rather explicitly a sequel series to Transformers Generation 1, though it takes place before and after it.
Transformers Armada was followed by Transformers Energon and (after some Executive Meddling to try and make it fit into the timeline) Transformers Cybertron. Collectively they later became known as the Unicron Trilogy.
Avatar: The Last Airbender is followed by The Legend of Korra.
Batman Beyond was a sequel to the entire DCAU most particularly Batman the Animated Series taking place in the nearly-distant future, where Bruce Wayne has retired and a new Batman takes up the mantle. Before that, The New Batman Adventures took place a few years after the original series, although it's sometimes considered just a continuation, not a whole new series.
Ben 10 was followed by Ben 10 Alien Force, then by Ben 10 Ultimate Alien.
Nicktoons Network's series Voltron Force is this to the original series from several decades ago, though only 5 to 7 years have passed in-story.
- Though its unclear if Voltron: The Third Dimension is still canon (at least in part) or not.
Extreme Ghostbusters is a sequel series to The Real Ghostbusters, with Egon mentoring a new team made of college students.
Speed Racer: The Next Generation, taking place 40 years after the events in the 1967 anime.
In-Universe example in Harley Quinn, where Clayface is trying to start a sequel series to Fuller House (the failed sequel series to Full House) called Fullest House.