Can I see what a person who typed in a specific search word did on my site?

Yes, in some cases. Login to your account, and go to Visitors: Visitors by search term.
Here you will see all the recent visitors to your site that used a search term: both the search engine, and if available the search phrase typed in. Note that in a majority of cases the search term is encrypted by the search engine.

By clicking on the green or red puppets you will be taken directly to their click-stream(s).

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