I know what speed runs are, but what does 'any %' mean when speedrunning 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)
"Any%" for any game is generally defined as "reach the end credits screen without using cheat codes." Each community, such as Twin Galaxies, speedrun.com, Speed Runs Live, TAS videos, Speed Demo Archives, Awesome Games Done Quick, et al, will define their own set of rules, with at least TAS being a weird outlier compared to other communities.
For example, TAS (Tool Assisted Speedruns) allows some console commands, input-scripting tools for mouse and keyboard input, segmented speed runs (recording each section of the game over and over again for best results), any game glitches, and frame-counted timings so the run doesn't have to be played at 100% speed. Notably, TAS does not allow New Game+ content, if any such content is applicable to a game.
Other communities may only count RTA (Real Time Attacks), not segmented runs, may allow NG+ content or mods, or may allow (or not) certain console commands, glitches, frame rate limits, and more. There may also be different categories, such as "any%", "any% glitchless" (no intentional glitching allowed), etc, even within a single community.
In addition, each difficulty level is considered its own "any%" category, such as "any% normal" versus "any% hell". This means that you'll find different times for different difficulties, and all of them may be simultaneously a world record, despite having wildly different times.
In other words, there is not precisely one "any%" for Diablo 2, but instead a couple of dozen various types based on community rules and different categories within each community. The TAS run is only 22 minutes long, while most RTAs appear to be around the 1 hour, 12 minute mark. They are all perfectly valid in their own community's rules, even if they wouldn't be accepted elsewhere.
That's the beauty of speed runs. There's something out there for almost everyone, from "casual" RTA to hypercompetitive TAS, and everything in between. Some communities might even allow "joke%" runs, where the objective is to reach some arbitrary condition (usually not completion, but something absurd like "collect 100 buckets").
Also, there's tangentially related speed runs, like "low%", where every item you pick up, even currency, can count against you; the goal is to reach the end credits with the absolute minimum amount of pickups. These are interesting because a new, longer run can be the record if you eliminate a single item from a faster, but heavier, run.
So, in conclusion, what is "any%"? It's whatever the community you're playing with, or watching, agrees that it is, and there are different, conflicting versions of what counts for an any% run. It's up to you to decide what you consider a legitimate any% run community. You may also find people who speed run by their own rules and are not necessarily part of a community, perhaps because nobody else wants to play by those rules. Speed runs are not about notoriety, so much as it is about having fun. You get to decide what counts as fun or not.
The "Any %" category means that the final score of the speedrun is judged entirely based on the finishing time regardless of what was done. This is in contrast to a "Low %" speed run, where a low % actually trumps a lower speed.
This video explains why the low% is so long, and it involves doing something really bizarre for many hours to skip a significant part of the game.