This post refers to using Twitter on the desktop Twitter.com version - things may well be quite different on Twitter for Smartphone apps and other apps (eg Echofon, which I use).
Yesterday I spotted, while pruning spam followers, that Twitter's added some extra options to the block function, and that I also have the additional option of muting people.
If you mute someone then their tweets don't show up in your timeline but they do show up in your mentions. This means that people probably won't know you've muted them, because their messages to you still get through. You can also find their other messages by visiting their profile page or searching for messages they've sent.
I've used my test account to mute myself. In the screenshot above you'd click on the snowflake / gear wheel icon and choose 'Mute this user' from the options. Once done you'll see a red icon of a muted speaker.
I block a lot of spam followers or people who've followed me for what I think are irrelevant reasons (because I once mentioned a word in a tweet that they're interested in). You used to be able to block 'without blame' but now Twitter requires you to fill in one of four options before you can block them, I don't really like this but that's what's changed.
What happens if you block someone on Twitter? What happens if they block you? (4 Jan 2014) - although the pictures are now slightly out of date the functionality hasn't changed.
Answers to common search questions
I'll put them here when I have a look at my Google Analytics logs - don't fret I can't tell who searched for what, meanwhile here's what Google suggests in its autofill.
1. Can I tell if someone's muted me?
1. How do you know if someone's muted you on Twitter?
No. To the best of my knowledge this information is not available to users (though presumably Twitter must know, obviously, so that it can avoid delivering the tweets). See also answer to 2.
2. If I mute someone on Twitter will they know?
No. See answer to 1. Additionally I am not aware of any notification sent when someone mutes someone else, I don't think Twitter wants to make this information available.
If you visit their profile page you'll see the red 'silenced speaker' icon (see first picture) so you'll be able to find out if you've muted someone (assuming you forgot). I suppose the only way they'd ever know is if they played around on your computer while you were logged in to Twitter, and visited their own profile (!), or you sent them a screenshot ;-)
3. Who muted me on Twitter?
It's not possible to tell.