How can I track the keywords that the users are typing in the search query mechanism to search my website ?
If you already have Google Analytics Code on your website then you could see all the keywords, visitors used to reach your site in the Traffic sources report. Under the Traffic Sources section there is an inbuilt report for keywords.
Install the Google Analytics tracking code on to your website pages, the information on how to do it is here: http://www.google.com/support/analytics/bin/answer.py?answer=55603, post this the Google Analytics tool will help to get the keywords, source of traffic, regions from where u get traffic and other relevant metrics.
This option is available in Google Webmaster Tools. If some one type a query and against that your website shows in search result, that will be an imppression and you can check this from your Google webmaster account. Also you can set custom reports for this purpose in Google Analytics.
Since the middle of last week I have seen a 200%-300% increase in traffic, almost all of it is spammer creating accounts.
That is one of the features that is provided by Google Analytics. You can see all of the keywords that are used by visitors when they come to your sites.
that easy !!!! you can click on the secondary dimension then click on traffic source and click keywords. It will list down all of the keywords that your traffic look for and reach to you . You may found some new keywords that you may target before but yet lot of people do search for it.