Personalized search is showing different results to different people on the same search. Rather than just using the keywords provided for a search, everything the search engine knows about a person is brought to bear and impacts the search results.
The goal is to provide more relevant search results. Since different people have different interpretations of what is relevant, at some point, the only way to further improve the quality of search results will be to show different people different results. So, the advantage of personalized search is that it promises to deliver better search results. The difference is likely to be particularly substantial when a search is very ambiguous (e.g. a single word like "desk") or someone has difficulty finding something and refines a search repeatedly.
What are the disadvantages?
First, privacy is an issue. Search engines would build a profile of everything you tell them about your interests, every search you've done, and every search result you ever clicked on. That's a lot of information and, unless handled in the strictest confidence, could make many people nervous.
Second, search results will no longer be consistent for the same search, certainly not for searches by different users and not even for many searches by the same user. This means that e-mailing a search to someone or easily finding something by searching for it again become a bit more difficult.
Third, personalized search is computationally expensive. Caching and many other optimizations become impossible when every search result list is customized in real-time.
Will the advantages outweigh the disadvantages? Big players are betting yes. Google, Microsoft, A9, and Yahoo are all testing personalized search.