They're still rolling out the new search box in twitter; not all accounts see it yet. But you can access by going to the bottom of your profile or home page and clicking on the link for search.
Many topics are denoted with hash marks, so you can search like that - or you can just search for a string. For example, one of the themes I use for a lot of my WordPress installs is called thesis, and the designer plus a good portion of the community is on twitter, so if I'm looking for something specific, I can type #thesiswp into that search and get all those tweets. Or I can just type netmeg and see all my tweets, plus all those that either mention or are directed at me.
(warning - don't follow me on twitter unless you are prepared for a *lot* of tweets, and some amount of cursing. I'm not near as polite as I am here, and I rant a lot. that said - I'm at
http://www.twitter.com/netmeg of course)
On WMW there's a lot of discussion these days about whether Twitter can replace Google as search engine. It definitely is an interesting discussion; I don't think it's too likely to actually *replace* Google. I've seen it referred to as a 'recommendation engine' and I like that term a lot more - you can get instant feedback on everything to parsing out an error message in your software to finding a good Thai restaurant in an unfamiliar city, often times from people you know or trust at least marginally - you can't really get that from Google so much. And it's fast.
Not to mention all the companies that are jumping on board to manage their reputations or offer customer service - I've had friends with Comcast issues who got them solved by @comcastcares, and I myself had a problem with a $160 rebate for Quickbooks that got refused - I bitched about it on Twitter and within half an hour a rep from Intuit (who was not previously following me) sent me a message asking if she could help. She found me through twitter's search, and before my problem had a chance to spread too far, she got in touch. I got my rebate.
Not to mention I have at least six Google employees following me now (two 'officially' and four under the table) so when I have an Adwords or AdSense or other Google problem, all I gotta do is start yelling about it on twitter and the messages and emails start coming in. Yahoo and MSN are on now too.
Twitter is definitely on Google's radar though; Google wants to be the one collecting and organizing ALL information, and think how much information is going through Twitter that Google doesn't have its sweaty paws on? And it's growing like crazy. From what I've read, Twitter had 500,000 users in January of 2008, and six million users in January of 2009.