Not sure exactly what you mean. Out of the box, from a user-perspective, you don't get the 'saved searches' feature that is available in other search engines. But you can configure either Alerts or RSS to revisit previous searches and see if any new content has been indexed since you last searched. Similarly, if there is a specific result you want to track (such as a document), using alerts or RSS means you will automatically be notified if the document is modified.
From an admin-perspective, you can track search queries and reports to find out what are the most/least popular queries and see stats like click-through rates, searches returning zero results etc.. This stuff is all useful for tweaking and improving relevance - you can use the info to determine whether or not Search Keywords (best bets and editorial text) may help, or whether certain sites need to be promoted (or demoted) as authoritative (or useless). It all helps tweak relevance ranking.
p.s. if you're not sure what alerts and RSS are all about, check out my blog post (get asked it quite a lot by customers) - http://www.joiningdots.net/blog/2006/09/sharepoint-and-rss-vs-alerts.html and to configure the RSS view web part - http://www.joiningdots.net/blog/2007/08/sharepoint-rss-viewer-web-part.html