(written from a Production point of view)
Story arcs are story lines in Star Trek that are told over the course of multiple episodes. They are not simply two-parters or recurring characters and themes, but rather plots that are interwoven with other arcs and plot threads. An example of this is Star Trek: Deep Space Nine's introduction of the Dominion and the eventual Dominion War. Story arcs often take precedence over other plot elements and consume a series for several consecutive episodes, but in many cases (such as the Dominion War) they may take a back seat and re-enter the picture later on.
Prior to DS9, story arcs were a relatively minor part of the Star Trek universe. Star Trek: The Original Series and Star Trek: The Next Generation both largely restored the status quo ante at the end of each episode, and even two-part episodes were somewhat rare. This changed due to the stationary nature of space station Deep Space 9, which did not simply move on to the next adventure each week; characters' actions had lasting consequences, and the events of one episode directly influenced the next. At the time, Ira Steven Behr had to fight with Viacom over making the show so serialized. 
Star Trek: Voyager took the idea of the story arc in a different direction by setting itself aboard a starship as per the traditional Trek formula but stranding its crew in the distant Delta Quadrant. This both enhanced and reduced the prevalence of story arcs; while alien races such as the Kazon and Borg appeared in multiple episodes, the show was hampered by its very nature: the object was to leave familiar elements behind, which meant each plot thread had a limited life span.
Star Trek: Enterprise, like DS9 before it, often relied on story arcs such as the Temporal Cold War and the Xindi arc that took up the entire third season. By the show's fourth season, Enterprise wrapped up these story arcs and instead refocused on the show's prequel concept with a series of "mini arcs." Most season four episodes are two- or three-part story lines, with few stand-alone episodes.
These episodes are originally broadcast in two hours, as opposed to the standard one hour. After their original run, feature length episodes are re-cut as two part episodes (see below), but are usually released in their original broadcast format as a single episode. They include:
These episodes air separately, but tell the same story. It is typical for the first part to end with the line "To Be Continued...". They are frequently used to end and begin seasons. Two-part episodes include:
- "The Best of Both Worlds" and "The Best of Both Worlds, Part II"
- "Redemption" and "Redemption II"
- "Unification I" and "Unification II"
- "Time's Arrow" and "Time's Arrow, Part II"
- "Chain of Command, Part I" and "Chain of Command, Part II"
- "Birthright, Part I" and "Birthright, Part II"
- "Descent" and "Descent, Part II"
- "Gambit, Part I" and "Gambit, Part II"
- "The Maquis, Part I" and "The Maquis, Part II"
- "The Search, Part I" and "The Search, Part II" (see below)
- "Past Tense, Part I" and "Past Tense, Part II"
- "Improbable Cause" and "The Die is Cast"
- "Homefront" and "Paradise Lost"
- "In Purgatory's Shadow" and "By Inferno's Light"
- "Favor the Bold" and "Sacrifice of Angels" (see below)
- "Image in the Sand" and "Shadows and Symbols" (see below)
- "Basics, Part I" and "Basics, Part II"
- "Future's End" and "Future's End, Part II"
- "Scorpion" and "Scorpion, Part II"
- "Year of Hell" and "Year of Hell, Part II"
- "The Killing Game" and "The Killing Game, Part II"
- "Equinox" and "Equinox, Part II"
- "Unimatrix Zero" and "Unimatrix Zero, Part II"
- "Workforce" and "Workforce, Part II"
Three consecutive episodes that air separately, but tell the same story.
Multi-episode arcs tell a continuing story and are spread out through several different episodes throughout one or multiple seasons.
The Next Generation
- Q and the Enterprise
- Worf vs. Duras
Deep Space Nine
- Changeling infiltration
- Klingon War
- Eddington vs. Sisko
- Dominion Invasion
- The Final Chapter
- Voyager vs. the Kazon
- Q Civil War
- The Hirogen's Hunt
- Barclay and the Pathfinder Project
- Temporal Cold War
- Vulcan vs. Andoria
- Journey to Risa
- Archer's trial by the Klingons
- Xindi War
- "The Expanse"
- "The Xindi"
- "Anomaly (ENT)"
- "The Shipment"
- "North Star"
- "Carpenter Street"
- "Chosen Realm"
- "Proving Ground"
- "Doctor's Orders"
- "Azati Prime"
- "The Forgotten"
- "The Council"
- "Zero Hour"
- "Storm Front"
- "Storm Front, Part II"
- Federation-Klingon War
- The Red Angel
- The Burn
- The DMA
- Synth mystery
- Pakled conflict
- Spock's death and resurrection, and the consequences thereof
22nd century episodes
23rd century episodes
24th century episodes
The mirror universe
The alternate reality
The Original Series
- The Talosians
- Polywater intoxication
- Shore Leave Planet
- Khan Noonien Singh
- The USS Defiant
The Next Generation
- DaiMon Bok and his revenge attempts on Picard
- The Traveler and his interactions with Wesley Crusher
- The Crystalline Entity
- Alien neural parasites infiltrate Starfleet Command
- Professor James Moriarty
- Arridor and Kol
- Vash and her relationship with Jean-Luc Picard and Q
- Thomas Riker
- Riker and the Pegasus
- Borg and First Contact (the Borg's involvement with Earth's first contact in 2063 and the repercussions which followed)
- Vulcan/Romulan Reunification
Deep Space Nine
- Anti-alien sentiments on Bajor
- Bajor's entry into the Federation
- The Maquis' struggle for self-determination
- The Dominion's threat to the Alpha and Beta Quadrants
- Kira and Odo
- Kira and Dukat
- Quark and Grilka
- O'Brien and Liam Bilby
- Dominion cold war
- Nog and AR-558
- The Hirogen and holographic technology
- The Silver Blood
- The Q Continuum
- Fair Haven
- Augments and Khan Noonien Singh
- Xenophobic Humans
- Harry Mudd
- Rise of the Kelpiens
- Tribble troubles
- The Conclave of Eight attacks Mars (prologue flashbacks in chronological order)