A Spectre (in American, Specter) is the Undead spirit which haunts those bourgeoisie as haven't yet accepted the reality of class-dialectic and the inevitability of Communism.
Dungeons & Dragons
In Dungeons & Dragons, the Spectre has historically taken the European spelling; 4e-5e has gone more 'mrrrken, for whatever reason. However you spell it this is just a powerful Wraith with twice the energy-drain. As late as 1977, Eric Holmes acknowledged that this was the ring-wraith, the "Nazgul of Tolkien" - i.e., as with the Balrog, the Lawyers had told TSR to file off those serials. If a Spectre kills you, you become a weakened Spectre yourself - heavily implied, that your undead spirit takes Wraith stats.
It arose in the first Monster Manual and the Eric Holmes 1977 box-set: the one bundled with B1, rather than with B2. In those ancient days, Holmes implied you could turn them on an 11-12 roll on 2d6 (=1 in 12, yay math) at fifth level; AD&D's "Matrix for Clerics Affecting Undead" (DMG matrix "III.") agreed, but boosting to 20 on a d20.
The Spectre will fuck up your party at 6th level, so one of them is a boss monster in X3: Curse of Xanathon. At higher level, the Restoration spell takes the edge off. At 6th (rather, 4th) the local priests can probably hook you up if you'd got the damage in the course of doing good - completing X3 assuredly counts.
How a Spectre doesn't just sail into town one night and make a Wraithopolis under his command is a problem that must be solved by any worldbuilder. Having them all as Nazgûl subject to a Dark Lord's command, which cannot exist independently, was Tolkien's solution (also the Nazgûl could only infect someone else with a Morgûl Blade which you can't exactly order on Amazon). In the Known World of the Expert series, that is more-or-less what one Spectre crew sets out to DO in one setpiece of X10.
The Spectre of DC Comics is a slain policeman who has been bonded to the embodiment of God’s vengeance that is tasked to punish the wicked with horrific deaths or far worse (since for some reason the Silver Age’s Comics Code Authority had a problem with death but not turning people into a fully aware inanimate object for eternity). Being an agent of Capital G God makes him perhaps the ultimate example of a hero so massively OP he’s difficult to write a compelling story for. The nearest thing he has to a weakness is being subject to brainwashing by wielders of divine artifacts and extremely powerful magic users (obviously far less common than kryptonite), which was made up during the 1970s to explain why he didn’t just kill the entire Axis leadership in a blink of an eye. As such he has been relegated to a plot device, rather than a main character, for most of his existence, despite being a very old part of the DC Universe (first appearance in 1940).
The Mayfair Games DC RPG gives him stats in Who's Who in the DC Universe #2 and regularlly uses him as an example of truely cosmic power throughout its many books. The DC license based Mutants & Masterminds books give him stats in in Heroes and Villains 2
World of Darkness
In the World of Darkness games [namely Wraith: The Oblivion], spectre was originally the term for a wraith/ghost of a deceased human who succumbed to Oblivion and became an agent thereof, but in the later Orpheus game the authors expanded the term to include sparagmoi spirits newly spawned directly from the Wyrm known as "Grandmother"; werewolves would refer to such Wyrm-spawn as Bane spirits.