Most iconically, the mechs of BattleTech are not the agile, energy sword-wielding, robot ninja mecha of the majority of anime. They are essentially futuristic tanks, lumbering weapon platforms that serve a specific tactical function. Though their imposing presence dominates the battlefields of the BattleTech universe, they are part of a larger military combined arms approach.
I. The Star League (2005-2780) - the alternate history of the BattleTech universe diverges just before this point, and this is the era of colonization of the stars and mankind's exodus from Earth (renamed Terra) following the discovery of FTL travel. One man eventually rises to rule over thousands of planets in the golden age of science and invention. An intergalactic empire is built, and then falls. This is the era where mechs are invented, and the political allegiances that would define the coming centuries are founded.
II. The Succession Wars (2781 - 3049) - After the fall of The Star League, there comics are period of almost undending war as the 5 great houses of the Inner Sphere struggle for dominance. The universe enters a technological dark age, as most of the advancements of the prior age are lost, and societies descend into a pseudo-feudal conditions. This is the default setting of the original BattleTech game.
III. Clan Invasion (3050 - 3061) - The Clans invade from deep space from beyond the Periphery, throwing the Inner Sphere into chaos. Their technology is far advanced, their mechs stronger, more adaptable. A common enemy brings the Inner Sphere together.
IV. Civil War (3062-3067) - A politically motivated unconventional war burning hot and cold that mainly involves two of the largest Inner Sphere Houses, though eventually almost all the Inner Sphere and some of the Clans are drawn into the conflict.
V. Jihad (3068-3080) - A splinter group of religious fanatics called the Word of Blake wages a war for technological superiority that threatens to destroy the newly-formed Star League from within.
VI. The Dark Ages (3081 - ) - The setting for the Mechwarrior Clix game. As the tabletop game hasn't caught up to this (they were originally under license not to touch it), I'll largely be ignoring this. The associated fiction is generally considered substandard.
I'm going to start from the beginning here, where the history of the BattleTech universe diverges from our own, and work my way through up to the Jihad. Obviously, after over 30 years of fiction and fluff, I won't be covering anything in detail, but just giving a broad overview for anyone new to the setting to gain a toehold. Most players tend to pick a favourite period and stick with it, and for me that's probably always going to be The Succession Wars.
After that I'll cover the game itself, the various options and resources available, the best sourcebooks and manuals, and good entry points for new players.