The search box is used for simple keyword searches. The box supports the new enhanced query syntax, which supports:
Properties search, like name: john
Phrases, like "john smith"
Simple term, like john
Include/exclude terms, like +john -smith
Prefix, including part of the word from the beginning (prefix exact matches)
Default AND between words (narrowing scope)
The out-of-the-box search box doesn’t support:
More complex logical searches, like (john AND Smith) OR (sharepoint AND search)
The search box is available as a configurable web part and is by default placed on the Search Center search pages. Furthermore all the build-in site definitions have the simple search box as part of the master page.