How Twitter messages work when you use @

First, imagine a fictional family where everyone follows each other on Twitter: @Mom, @Dad, @Brother and @Sister.
1. The basics

Rule #1: A person must follow you before you can DM them. If @Mom and @Dad (fictional example names) want to write completely privately so no one will see what they write but each other, they have to DM each other.

Rule #2: Anytime you put an @someone anywhere in a tweet, it will be sent to that @someone, regardless of whether they follow you or not.

2. Creating tweets only your "family" can see 
But what if @Mom and @Dad want to write tweets that @Sister and @Brother can both see, but no one else can see? Then they tweet like this:

@Dad Message goes here   -or- @Mom Message goes here

So, rule #3 is: When you start your message with @someone, the message will only be sent to @someone and the people that follow both you and @someone. Of course, anyone who follows both @Mom and @Dad will see these tweets, but for the purposes of our example let's assume only @Brother and @Sister both follow @Mom and @Dad.


3. Sharing "family" tweets with everyone So what happens when you put @someone after the beginning of your tweet? It is sent to all your followers, and to @someone. It's just a regular tweet, delivered to all your followers. Why do that? It's a way to share the conversation you're having with @someone with ALL your followers. If you're just saying something directly to them that you want everyone to see, it's common to write

.@someone message goes here

If you're just saying something about them you want them and all your followers to see, you'd tweet something like this:

So like I was saying to @someone the other other day...

But anything you put before you write @someone will have the same effect. So both of these messages are sent to all your followers and to @someone. It's just common to use a period "." if you're talking to them directly, but you want everyone to see it.

4. How Tweets are viewed, and how to make it work the way you want
Twitter applications and the Twitter website provide several ways to see your tweets. One way, called @mentions has a problem you might want to work around. This is typically how most interfaces show you tweets "sent" to you. The problem is that @mentions are a list of every tweet that has your @username in it. If lots of people you don't want to hear from start writing tweets with your @username in them, it can be overwhelming. This happens to popular/celebrity Twitter users, and also to people who get retweeted a lot. What to do? Simple: add a search for message sent just to you to your favorite interface. The Twitter search terms for messages that start out with @Twitter_Tips, for example, is:

to:Twitter_tips

Since the web interface will save searches for you, I have this one saved: a search for messages sent just to me.Here's how that works:

5. The exceptions   The exceptions are mainly just ways people can see everything you tweet, regardless of how you are using @'s

Exception #1:

"Who sees a tweet?" usually means "When I tweet something, who does Twitter send it to?" People can also see your tweets by going directly to your Twitter page. Everything you tweet is shown there to everyone (except your DMs). If you don't want that, you have to make your tweets private. The only time anyone will visit your Twitter page in most cases is the first time they hear about you. Then they'll just check out your page to see if you are the kind of person they want to follow.

Exception #2:

People can search for tweets, and all your tweets (except your DMs) that match what is being searched for will show up. So your tweets are more public than you might realize, especially because of...

Exception #3:

Some applications show you your followers what you tweet by using Twitter's search features. That means they'll see everything you tweet (except your DMs). Most applications don't work this way though.

Exception #4:

Ignore this one! It may no longer apply, or only very rarely apply. But for historical sake: When you click a "Reply" link in any interface, it writes the first part of the tweet for you, such as

@someone

...then you simply add what you want to say to @someone and everything works normally. But what happens if you don't click anything and just type (or cut and paste in)

@someone

It looks exactly the same, doesn't it? But Twitter could originally tell if you clicked a "Reply" link or not, and the rules only applied if you clicked the link. If you typed or cut and pasted the @someone in, your tweet was shown to everyone, same as if you used the .@someone trick.