You can block absolutely any user on Twitter by navigating to their profile (on the web and on mobile) and clicking the "User Actions" button located beside the Follow/Following button.

Blocking a user prevents them from being able to follow you from the blocked account. If the blocked user tries to follow you, they won't be able to do it, and Twitter will display a message that says "You have been blocked from following this account at the request of the user." 

Do Users Know When You've Blocked Them?

So, do they know? Well, that depends.

Twitter says that it does not send out notifications to users that have been blocked. The only way a user can tell for sure that they've been blocked is by visiting the profile of the user that blocked them, trying to click "Follow" and seeing the Twitter block message as written above.

Generally what might is that a user might notice that another's tweets or @mentions don't show up in their own timeline or interactions tab anymore. When they go to visit that user's profile and try to re-follow them, they find out that they've been blocked.

It's basically up to the blocked user to find out on their own if they've been blocked, by actually going to the user's profile who blocked them and trying to follow them again. If they don't even realize that the user who blocked them is missing from their timeline, they may never even know that they've been blocked.

Is There a Way to Prevent Someone from Finding out You've Blocked Them?

Unfortunately, there's no way to ensure that a user absolutely never finds out that they've been blocked.

The block message that Twitter displays will always be shown to users that try to re-follow an account they've been blocked from.

There is, however, something else you can consider doing. You can make your Twitter account private so you can avoid blocking people in the first place.

When your Twitter account is private, anyone who tries to follow you must be approved by you first. If you simply don't approve their follow request, you won't have to block them, and they won't be able to see any of your tweets either as an added bonus.

Keep in mind that the social web is a very open place, and managing your privacy settings is important if you don't want to be as open as the social web encourages you to be.

If you believe that a blocked user may also be considered a spammer, you can report the account to Twitter so that it may be considered for suspension.