The short answer is you can simply go to search.twitter.com and type in a word to see who is talking about it.
Depending on the word or phrase you type in, you may have to refine your query to get the results you are looking for and not be bombarded with new search results every second.
Here’s an example. You’ll notice in this search result for the term “tweetup” we get results for tweetups from all over, even Edinburgh.
What I’m really looking for is a tweetup I can go to in Los Angeles, so after refining my query I get this.
You can search Twitter for anything. The biggest use of search Twitter often is by people searching for their name or a brand monitoring their reputation online. Many businesses tweet so they monitor what people are saying about them so they can respond appropriately.
There are many apps such as, Twitt(url)y and TweetBeep, that help you track tweets with the terms you are looking for.