Google Analytics does record the IP address of your visitors, but it uses Cookies to "track" vistors.
If someone visited your site 10 times from a search results page from a search engine then they would be registered as 10 page views.
How many visits it would be registered as would depend on the gap between the cilcks. By default, the cookie that "times" a visit in Google Analytics expires after 30 minutes - but each new click (pageview) resets that timer...
What also happens is that the "source" and "keywords" for that visitor is update on each visit. So, if all 10 clicks where on the same day, the visitor would be attributed to the search term from the 10th (and last visit).
This is a good explanation of visits, visitors, page views, clicks:
http://www.google.com/support/analyticshelp/bin/answer.py?answer=1257084
this is a good explanation of the various cookies:
http://www.analyticsmarket.com/blog/google-analytics-cookies
and this is an explanation of the attribution of trafic sources:
http://beantin.se/post/420307062/traffic-sources-explained-understand-google-analytics