Pokemon Showdown is first and foremost a battle simulator, which means it lacks the story mode in favor of focusing 100 % on the battling aspect. PS can be used both in the browser or be downloaded in the computer, but there isn't much difference.
You can fully customize any and all Pokemon to any and all teams, regardless of ingame viability or availability, and all and any Pokemon are available right of the bat. The teams are saved in your cookies, not in the cloud or on your computer. This means any device that can have cookies can support PS, although it is far more optimized on PC.
Important to note is that there are several places you can do battles, so called servers, and servers can have different rooms for all sorts of businesses. Some are big and some are small, with quite different focuses and people visiting them. By far the biggest server is the Smogon University, the foremost authority on competitive Pokemon battling. This one server is big enough to have several rooms the size of dozens of smaller servers.
Your primary battling partners will be other players, as that is what competitive is all about. There is a ranking system in place to make sure you battle of the same rank as you, but these are tier specific, rather than linked to a single account.
There are dozens of formats to battle, but the most popular are the Smogon Tiers, but that's for another time. What PS have over ingame competitive battling is that it is quick to get going. In five minutes you can have full team up and running, compared to the days or even week's it would take to IV breed and train a team ingame. Is something not optimized, need to change a move? That takes seconds.
Here is a link to all servers and here is one to our own DB Server. If you have any other questions or wondering anyone there would be happy to lend a hand.