Witnesses are the people/servers who run the Steem Blockchain. Witnesses are responsible for setting some of the parameters of the blockchain, such as the price of creating new accounts, for example. The most important role is that we have to decide whether or not to adopt a hard fork for Steem, when they are proposed... So basically, witnesses - collectively - decide the rules of Steem and so also decide the future of Steem.

You can read more details about all of this in my original witness application thread.