SwagMeowth did a great job of explaining above, but I'll just elaborate on some stuff he missed and try explain all the main things
What is Pokemon Showdown?
Pokemon Showdown is an online Pokemon battle simulator. This means it allows you to battle other users from around the world in realtime. It's one of several Simulators, including Pokemon Online and Pokebattle. It's also the official simulator of the competitive community Smogon which unofficially governs most of the rulings in online Competitive Battling, as well as providing tips and strategies that can be implemented in both Wi-Fi and Online play.
What can I do on Pokemon Showdown?
Pokemon Showdown allows you to create Pokemon Teams from scratch, giving you the ability to choose the members of the team, their moves, EVs, IVs, items etc. without any restrictions, provided the actual sets are valid (ie. Spacial Rend Magikarp isn't allowed in most tiers).
How does Competitive Battling work on Pokemon Showdown?
You create your teams, and look for a battle in a specific tier. This puts you in a battle against another individual who is around your rating (You start at 1000 rating), and if you win you gain a certain amount of ELO, while losing makes you lose a certain amount of ELO (Pokemon Showdown uses the same rating system as Chess - ELO is the numerical value assigned as your rating). If you achieve a high enough ELO to reach the top 500 players of a specific tier on Pokemon Showdown, your name and rating will be displayed on Pokemon Showdown's Ladder as one of the top 500 players in that tier.
During specific times, Suspect Votes take place - this means that Smogon is considering to ban a certain Pokemon. During these votes, a new format is created (ie. OU Suspect instead of just regular OU) which eliminates the suspected Pokemon. This format is removed after the vote is over. The important rating in a Suspect ladder is not your ELO - it's your Glicko-1. Glicko-1 is Smogon's own little thing which allows them to factor in both the amount of battles you've won as well as your win-loss and all that. People who reach a certain Glicko are qualified to vote in the Suspect Vote.
Currently Mega-Metagross is being suspected from OU to Ubers
What are NU and all of those tier stuff?
There are a few main tiers on Pokemon Showdown, with dozens of other random ones as well. I'll list the main ones from highest to lowest and give a brief overview. Remember, any Pokemon from a lower tier can be used in a higher tier.
Anything Goes
This is the highest tier - only one Pokemon has been officially banned to this Tier, being Mega-Raqyauza. This means Mega-Rayquaza is the only Pokemon which can ONLY be used in Anything Goes. Any Pokemon can be used in Anything Goes
Ubers
The second highest-tier, and the highest one before the introduction of ORAS. Ubers contains all the Pokemon that are deemed too powerful for OU. Most of Ubers consist of the strongest Legendary Pokemon (Mewtwo, Arceus, Xerneas etc.) , with a few regular Megas alongside them.
Overused
Abbreviated to OU, this is where most of the battling happens on Pokemon Showdown. This is also true for Wi-Fi battles, as most Wi-Fi battlers also stick to OU Pokemon and lower. OU is comprised of a variety of fully evolved Pokemon and slightly weaker Legendaries (eg. Mew, Zapdos, Terrakion).
The next three tiers are UnderUsed (Also known as UU) and RarelyUsed (Also known as RU) and NeverUsed (Also known as NU)
PU
Pokemon that can't evolve any further but are just too horrible for any other tier are thrown into PU basically. Obviously, it still has stronger Pokemon that still can evolve, as do some of the other tiers including RU and NU. There is no real name for PU, though a lot of people say Poorly Used or Partially Used.
LC
All the Pokemon that have not yet evolved (they're in their basic stage) are in LC basically, with a few exceptions that are too strong and therefore banned from LC. LC has special restrictions, mainly in the fact all Pokemon in this tier must be Level 5.
And that's all the main Singles tiers.
VGC
VGC is a different format from all the ones above, which are Singles tiers. VGC is the official tier that is played in actual Pokemon tournaments across the world. VGC basically bans anything over 600 BST from memory, but an experienced player like AD or NP can correct me if I'm wrong :P. All Pokemon are also forced to Level 50, and you choose 4 Pokemon to participate in the battle from your team of 6. It's also played in a Doubles format.
Other Tiers
There are a variety of other tiers on Pokemon Showdown including and not limited to;
Random Battles (Showdown generates a random team for you based on their coding), Hackmons (Your Pokemon can basically have any ability, moveset etc.), Monotype (All Pokemon must share a single type on your team) and Smogon Doubles (Doubles using Smogon's own ruleset). There are also tiers in between OU and UU, UU and RU and so on - these are known as BL (Borderline), BL2 (Borderline 2) and so on. These tiers are restricted for Pokemon too strong for the tier below them, but don't receive enough usage to enter the tier above them. However these Pokemon can only be used in the tier above them or higher (Basically, there is no such thing as a Borderline tier Battle. Any Pokemon in BL can only be used in OU or higher.)
How are the tiers decided?
Singles tiers are decided first by usage, then by suspect votes and bans. If a Pokemon has enough usage, it moves up to a higher tier. If a Pokemon is deemed to possibly be too strong for a certain tier, generally a Suspect Vote is called and general community members strive to meet the requirements. If enough people vote to ban the Pokemon it is banned. In some cases, the "Council" of the Tier will vote between themselves and not open the voting to the larger community. The "Councils" of each tier generally comprise of some of the best players of that tier who actively participate in the community.
Other Links
If you want more you can check Smogon's FAQ here.
If you have the energy to and you can read coding, you can check Pokemon Showdown's Github which contains all of it's coding and take a look at the rulesets and everything.
Smogon's website and forum is also a helpful place.
If you decide to try out Pokemon Showdown and create a team, you can get it checked over and rated by a large amount of sites including Smogon's Forums, Serebii and of course, our very own Battle Subway. If you do decide to post on any of the above, please read their respective rules for posting teams beforehand, or your post will probably get hidden.
You can also ask around in Chat or something. If you have trouble, just look for an experienced player to give you a little advice. We also have a Apprentice-Tutor thread, with some of our site's best players offering to be Tutors. Some of the Tutors are busy however (Life sucks), and Timezones are problem so it might be difficult to find one depending on your circumstances and theirs.
If you have any other questions, you can give them a quick search on google, look through here, ask another question here or ask someone on their wall or via Chat.
TL:DR
Pokemon Showdown is where Pokemon nerds who get bored of Wi-Fi go to procrastinate and not do anything constructive.
Just kidding. :P
