Basically they're the ones that are running the blockchain (their servers). While centralized solutions are controlled by one entity that has full control over the corresponding servers, a blockchain is controlled by many. On Steem they're called witnesses, for Bitcoin they're called miners etc. They're also responsible to democratically decide on future changes of the blockchain called hardfork.
You can (and actually should) vote for your favourite witnesses which makes their selection being a democratic process. Before voting you should inform yourself well about the whole process and impact of your vote.
For further details you can check the FAQ for instance: https://steemit.com/faq.html#What_are_Steem_witnesses
I'd also recommend you to read this post from @lukestokes, currently one of the top 15 witnesses:
https://steemit.com/witness-category/@lukestokes/steemit-is-free-but-your-witness-vote-determines-its-future
Here you can see the ranking and vote for your favourites: https://steemit.com/~witnesses
I hope that helps to get an idea of it :-)
I'm not an expert of witnessing, so if there is someone who wants to add something I missed to mention, I'd be happy about that.
@surfermarly and now I understand. 😍