If you're following somebody and tweet from your account something like "@username hi something something" they then get notified that you've said something to them, so they will see what you've said. So they don't have to follow you per se for you to ask them something.
So use it as a form of communication and ignore the following thing if that's the bit you don't understand.
Search for "tile" or "tile adhesive" and add the firms you know of (not just all of them - that's the mistake I made and as a result I get spammed a lot every day - as somebody said above already).
If you keep doing @username messages to different people, eventually they'll follow you if they're interested in what you're saying and then you don't need to keep @username'ing them, as whatever you write to your 'wall' (facebook reference there to help you understand what that is) all your followers see.
So like neale did a message "anybody got xxxx tile", the tile shops following him would have spotted it. If he didn't have tile shops following him, he would have had to @tilegiant then the message (for example - TG don't have a twitter account by the way) to each of the tile shops he can find by searching.
To be fair, it is a bit weird and christ knows how something quite complicated in terms of communication got so popular, but it did.
You'll notice as soon as you have your account all the TV programs that have "#ProgramName" somewhere on the screen as certain points which means if you search twitter for the programname bit you'll end up with all the people who are talking about that program while it's on. You can then follow the 'interesting' people, or just keep searching when you want to see what's being said.
I'm not much for it, but as a web developer I suppose I have to have an account. Doesn't really get used much though.