For the purposes of this w/u, the only difference between a strategy wargame and a tactical wargame is the implied scale. In practice, they can be considered essentially the same.
Tactical RPGs, also occasionally known as strategy RPGs, blend the elements of traditional PC turn-based strategy games and console RPG character advancement. The best known example would be Final Fantasy Tactics, to the extent that the genre is often described as "games like Final Fantasy Tactics." (It's ironic that FFT was panned in Japan for being too similar to a previous title, Tactics Ogre, but this is a story for another node.
A tactical RPG is a war game where each unit is developed like a console RPG character. Generally, only games that balance the importance of the two evenly are considered tactical RPGs by purists, and generally only purists give a damn. The common elements shared by "pure" tactical RPGs include...
Chessboard-like battlefields, often with clearly delineated squares
Isometric camera views (only for games dating after X-Com: UFO Defense and Tactics Ogre; Shining Force and Fire Emblem use straight overhead views originally due to technical limitations)
Multiple classes for characters, attained by building up levels or some sort of specific kind of experience points. This can be as simple as a Final Fantasy I-style one-time promotion, found in Fire Emblem and Shining Force, to complex systems like Disgaea's reincarnation system or Final Fantasy Tactics's Job system.
Complex, heavily political stories, often loosely based on periods of history (largely the influence of Fire Emblem and Ogre Battle.)
Fantastic, usually high fantasy but sometimes science fiction setting (Historic strategy games, like Koei's Romance of the Three Kingdoms, tend to have more of a simulation nature, separating them from tactical RPGs.)
Emphasis on scripted, rather than random, battles, with the story is told during the battles themselves.
Abstract, instead of realistic, combat. As most tactical RPGs are console RPGs as well as being high fantasy, combat will generally include abstractions like discrete turns and hit points.
Unusually high levels of difficulty, particularly with regard to mortality. Death tends to be permanent, or at least very, very difficult to reverse. This is less common in later titles, but will generally be found in most tactical RPGs before 2000.
Not all of these elements are always necessarily present, but a game with more excluded than included will often be pidgeonholed into another genre.
While few agree on a strict definition of tactical RPG, the flagship games of the genre are widely agreed upon.
Fire Emblem, Nintendo's (up-until-recently) Japan-only series on the NES, SNES, and GBA, is one of the earliest and longest-running series in the genre. It is also one of the least well-known outside of Japan, as Nintendo has done little to promote it.
Shining Force, developed by Camelot and published by Sega on the Genesis/Megadrive, Sega CD, and Saturn, has been absent since Shining Force III. The first installment in the series is moderately well-known, as it has been included in almost every version of the Sega Smash Pack, a compilation of Genesis games and remade as Shining Force: Resurrection of the Dark Dragon on the Game Boy Advance.
Ogre Battle/Tactics Ogre, developed by Quest and published by Atlus and Nintendo at varying times on the various systems, influenced many later games, but has declined since the defection of Yasumi Matsuda to Square. The future of this series is murky, as Square bought Quest out.
X-Com, a PC-only series originally developed by Mythos Games, with sequels and publishing from Microprose, isn't a console RPG, but has much in common with other tactical RPGs.
Final Fantasy Tactics, developed and published by Square on the PSX, was directed by Yasumi Matsuda after his defection from Quest. A cult hit when originally released in the US, it became the best-known game in the genre, especially after a Greatest Hits rerelease due only to popular demand. Unfortunately, its legendary difficulty and infamously incoherant translation makes it something of an acquired taste, even for genre fans.
These aren't the only games widely considered to be tactical RPGs; lesser-known games like Hoshigama, Disgaea, and the Front Mission series are simply not as well-known or well-liked.
As of this noding, tactical RPGs are in a bit of an upswing. Both Final Fantasy Tactics Advance and Disgaea have met with positive reviews and decent sales (the former partially because of a major promotional campaign on Nintendo's part), but Gladius, a fantasy/gladiator title coming from LucasArts, has had rather lukewarm previews. Nintendo, in a bit of a surprise, also recently announced that the next Fire Emblem title is going to be coming to the US.
With the proliferation of "RPG elements," which generally means levels of experience and equipment, the loose definition includes many, many different games in the last several years, including games which arrived at this point through parallel evolution. (Warcraft III, for example, is a real-time strategy wargame with RPG elements grafted on, rather than a hybrid.) Many strategy and tactical wargames have had RPG-style character development added to help increase depth and foster attachment to characters, and many RPGs have added increasingly more tactical combat to liven up otherwise boring random battles. Notable recent maybes are Suikoden III, Arc the Lad: Twilight of the Spirits, and Warcraft III.