DDOS (Distributed Denial-of-Service) attacks are a sub category of malicious computer network interferences called DOS (Denial-of-Service) attacks. As the name suggests, Denial-of-Service attacks basically mean interference, slowing down or in some cases completely breaking a host or server's access to the internet. DDOS attacks are a special category of them, where multiple computers, remotely controlled by the hacker, are responsible for bringing a large scale computer network problem, resulting mainly in crashing of large web servers, insecurity of data and remote access.
Imagine a computer network of say 10 computers connected in a LAN which is connected to the internet. In a DDOS attack on one of those computers, a remote computer (the hacker) would use several unsuspecting machines which are compromised using malicious Trojans, to facilitate sending of internet packages rapidly to the victim computer, causing a server overflow, which may either slow down the access to the internet or even take over the entire bandwidth of the LAN, causing the internet access of the entire LAN (10 computers) to break down. On a large scale, this approach of flooding the bandwidth has led to the falling of various banks and other large-scale websites. The tracing of the culprit, is difficult due to the sending of packages through unsuspecting machines (called botnets), scattered randomly across the world.
You can know if you have been DDOS attacked, when your access to the internet is absurdly restricted, and in some cases, a huge amount of spam mails received (called an e-mail bomb). The sad part is, it is very difficult to escape one, and sometimes you may not even know if your PC has been a botnet. For a day to day home-based, personal user, it is safe to check on malicious and infected data entering the PC regularly with proper internet security, firewalls and antiviruses, though these are highly ineffective too. DDOS attacks are illegal in most jurisdictions of the world.