What happens?
Customers: they get a shutdown notice and the service stops working for them, and have to find competitors that do the same thing.
Founders & employees: laid off and go look for jobs.
Board members: move on with their lives.
Data, IP, office furniture, leftover inventory, rubber bands: sold at auction to pay creditors.
Social media accounts & domain names: technically this is property of the corporation and is also sold at auction, but in a sale that assumes no acquisition offer for team/brand/IP, they're often worthless. They become zombies and eventually revert to the public under whatever inactivity policy the host has (eg. domain names often get snapped up by speculators).
Debt: There's a pecking order for debts and other capital claims, and whatever money left in the company checking account + the liquidation sale mentioned beforehand goes to it. IIUC, unpaid wages come first, then unpaid suppliers. Then senior debt with liquidation rights negotiated into the contract (eg. my wife invests with philanthropic debt, which often has these clauses). Then regular debt, bondholders, then preferred stock, then common stock. Founders & employees usually hold common, which is the last to get paid out.
All the startups I've personally shut down did so with no drama, but there is a big tendency to fight over the carcass, particularly when you have people with bigger egos involved. One that I worked at, for example, involved an IP auction where the founder stealth-bid (through an employee friend) against the VCs, the auction house accidentally told both parties they had won, the lawsuits went flying, and the founder moved to China with the IP to start the exact same company and run it into the ground again.
When the company files for bankruptcy, the court appoints a bankruptcy trustee (Google this for more information). These are public employees of the DoJ, paid for by tax dollars. It's their responsibility to ensure that the assets get liquidated and the cash is distributed properly to creditors.
They will usually contract with one of the liquidation specialists mentioned above to handle the actual logistics of selling the assets. I'm not sure exactly how the liquidation house gets paid - I'd guess they take a commission on the sales. During recessions or bubble-bursts, you can often get great deals on office furniture and computers by looking for bankruptcy auction notifications. (In the absconding-to-China startup I mentioned, my boss called me up and said "Hey, if you want really cheap furniture, there's a liquidation auction going on in a couple weeks", but I was off at college at the time and didn't have space in my dorm room.)
This is all assuming you can't work something out amicably with all the parties involved. In many tiny tech startups - the ones that are a couple friends working out of their apartments with no investors - the founders just agree to put in job references for each other, split the company bank account evenly, let the IP & domain name die, and stop paying for any cloud services they're using. Sometimes if you have investors but the company clearly isn't going anywhere, you can arrange to return the remaining cash pro-rata and the investors will either fund your next venture or land you a job at one of their portfolio companies (Evan Williams famously did this with Odeo, and his next venture was Twitter.) You can avoid bankruptcy court and the whole liquidation process this way, you just need to have all your creditors & investors on board.
That's one example of a worst case scenario.