The search box next to the NEW ticket tab performs system-wide full-text searches in a Google-like interface. Search results include tickets and topics. 

Searching in Zendesk works as you would expect; you enter words you wish to search for, and Zendesk returns all items that contain ALL of the words entered. The result set is ordered intelligently, e.g. tag matches are ranked higher than content matches.

In addition, Smart Search offers various operators and property keywords. With Smart Search you can combine your free text search queries with smart queries and perform searches such as "HP driver assignee:me" and "priority>normal and not updated:yesterday". Embracing Smart Search can speed up your day-to-day work routines. You can read more about smart searching here: https://support.zendesk.com/forums/3199/entries/3152.

The search box can also be used to lookup users, simply type e.g. "user:john" if you want to lookup a user named John.

Finally the search box also serves a shortcut provider for looking up tickets. Simply type "#233" to go directly ticket number 233.