As I tell many of our clients, your tweets aren't just seen by your followers! Unless you've 'protected your tweets,' your posts also go into what's called the "public timeline." It contains all the tweets that are being pumped out through Twitter by people like you.
I like to think of the public timeline is like a giant bowl of M&M's. If you want to eat only the blue M&M's, you can easily pick them out of the bowl.
Twitter Search does just that.It allows you to filter the public timeline and find people that share the things your interested in. Perhaps you are on a "vegitarian diet."
In addition, there are some-thousands of Twitter applications that also integrate the public timeline and search/filter capabilities. You can see how valuable it is that your tweets be included in those public spaces.
In the name of spam protection, Twitter has sophisticated methods to identify a spaming twitter account. If you've been flagged you tweets will not be seen in the public timeline (oh no!).
To check, search twitter with your twitter handle (@yournamehere)
If you saw your tweets appear in the search results your okay. If you didn't see your tweets, check to see when you tweeted last, if it's been a while try a test tweet and see if they show up. If they still aren't showing up, then you need to fill out a support ticket with Twitter. Here is the form to do that
Hope this helps!