The easiest and most direct way to search Twitter is through the Twitter Search page located at http://search.twitter.com. Not everyone is aware of it, but Twitter has long had a special page set up just for tracking tweets.
As you can see, it looks a lot like Google's home page. If all you want to do is create a simple search, you can just type in your term and hit the search button. If you want to get a little bit more complex, hit the "Advanced Search" button.
Search Twitter With Advanced Search
Advanced search is perfect if you want to search for an exact phrase or make sure the search results skip anything with a certain word. For example, if you want to get the latest news on the Dallas Cowboys, you can put that exact phrase in the box labeled "All of these words". However, if you want to get news about Dallas but nothing to do with the Cowboys, Stars or Mavericks, you could put "Dallas" as your search term and in the textbox for "None of these words" you can list out those team names.
The main search page also contains trending topics. These are topics that are currently very popular. Unfortunately, these trends are very susceptible to spam attacks, so their use is marginalized.