It's a method of sending so much data to one computer / IP Address, usually, but not always, a simple "ping", that it essentially gets flooded, can't handle it, and either gets completely knocked offline, or becomes so slow that for all practical purposes that computer, or even website, becomes unusable.
Back around the late 1990's when I was in the "cracking scene" is was just "DOS" (not distributed). In those ancient times your average user was on a dial-up modem (9600, 14.4K, 56K, etc.), not broadband like today, and you'd usually get a new IP address every time you logged off and then back on again (by dialing into your ISP each time). The speed of those modems were dog slow, and they couldn't handle much traffic, in fact, for anyone who wasn't there to witness them, to give you an idea of how slow they were, an average sized mp3 file that you can download today in around a minute or less, could take an hour or more to finally download, assuming all went well and there were no hiccups during the transfer and/or you didn't get bumped offline by an incoming phone call, so "distribution" while DOS'ing wasn't even needed, you could just use the plain ol' ping.exe in Windows or the Linux version of ping and set it to just keep on pinging until it flooded the modem and knocked the person offline.
And it happened A LOT, especially to us IRC'ers, surrounded by bored college and/or high school kids. Hell, who am I kidding, I used to do it sometimes too, it was both fun and funny sometimes.
... and broadband modems that can take so much incoming data that a single computer sending pings just won't do the trick. Enter the "Distributed" portion of the name. The idea behind the DDoS is the same, however, instead of a single computer flooding the target computer with data, many computers all send it at once, achieving the same end result. The one target computer cannot handle the constant flow of incoming data from so many computers at one time. Eventually it's unable to reply to pings, or respond at all, resulting in becoming unusable, and if any services, like a web server, or FTP server as just 2 examples, are running on that computer, they'll also be unusable. Anyone trying to connect to any websites running on that computer's web server won't be able to connect to it, it'll just time-out.
The usual way is by infected computers. For example, let's say you unknowingly run some non-photo attachment in one of your emails (never open attachments from anyone you don't know! And still, be cautious of attachments from those that you DO know because they might not even know it's infected), the attachment secretly installs software on your system that can allow someone you don't know (the attacker who wants to do a DDoS) to run certain commands, such as ping.exe, and often do anything that you can on your own computer, all without you even knowing it's happening. You computer is now what they call a "zombie"; it's just sitting there, infected, waiting for the command to awaken and attack. Now imagine a person who has access to hundreds, even thousands of infected "zombies", sending the command for all of them to flood a single computer with pings, or some other data, and you can see how easily a computer can be crippled by the attack.
Most of the time this network of "zombies" (can also be called a "botnet") isn't controlled by a single person, but rather a group of people who have access to this "botnet". So more than one person can send DDoS commands whenever they want to, often via IRC.
The standard rules apply:
Even with all the precautions you can possibly take you still just may end up getting bitten once in a while simply because there are hundreds, possibly thousands of people out there working diligently day and night to find exploits in software and web browsers that will allow their payload to sneak onto your system, but with the above tips and just some awareness and caution you should be quite safe. Over the last 17 years I've only been infected 2-3 times and my internet travels have led me to the shadiest of places .