Mining is the process by which cryptocoins comes into existence.
It also allows to maintain the blockchain's integrity by checking each transaction and constructing the public ledger.
You can do it with your own household computer, buy expensive components and install a mining software.
However, the power needed to produce a single coin is increasing every 10 minutes and virtually no one can do it on any cryptocoin (be it Bitcoin or Litecoin, Darkcoin, Dogecoin, ...) by himself.
Miners have regrouped into pools to gather their power together and be a lot more efficient.
Companies created computers (ASIC miners) designed to perform only the simple task of hashing sha-256 (Bitcoin), Scrypt (Alt-coins including Litecoin, Dogecoin, and a lot of others) and soon new algorithms such as x11 (Darkcoin, ...).
Those computers are becoming harder and harder to come by and will no longer be efficient unless you buy the biggest of those.
That's when Cloud mining came to being