What is an any % speed run in Diablo 2?

While I haven't played Diablo 2, I can tell you any% speedruns are speedruns with the goal of finishing the game as fast as possible, while allowing the player to ignore side content. The percent symbol refers to the % of activities completed in the game - in this case it refers to not completing side activities in the speedrun. Other speedrunners attempt to complete all activities available in the game, known as a 100% Speedrun.

You can see the rules and descriptions specifically for diablo 2 here https://www.speedrun.com/d2lod (press 'view rules' button)

Any% in d2 means that the timer starts on character creation of a lvl 1 Char in Act 1 and ends when Baal dies within the category difficulty (normal, nightmare, hell).

Generally, as @PausePause has said, Any% (without any qualifiers) involves using "any means necessary" to "complete the game" as quickly as possible - the quotations are there because what exactly those terms entail vary from game to game, and is generally agreed upon by the speed-running community of the individual game. The terms for Diablo 2 (As @jstq has provided) is a relatively uncontroversial example.

Typically, "completing the game" means defeating the final boss or reaching the end credits screen, and "any means necessary" allows for any glitches, bugs and exploits (AFAIK cheat codes are generally, but not always, disallowed).

This very permissive definition often allows for strange exploits that reach the end screen while skipping pretty much the entire game to qualify - search for "Super Mario Bros. 3 any%" for an example. Diablo 2 however does not seem to be one of those cases.

Also, do note that any% does not ban players from completing side activities - in fact, quite frequently speedrunners will go through technically unnecessary content if time spent playing said content can be offset elsewhere by whatever it unlocks (e.g. obtaining a fast vehicle)