Simple search is an input box in the upper right of all pages, there is "Search" button next to the box. There is "Simple Search" button in the menu, as well, that leads to the simple search page with some hints.
Simple search is magical in some way and you may find this magic cool or annoying (that depends).
Just type search terms (keywords) in the text box and hit enter or click Search button. The search will be done so that the result matches each keyword (AND).
Magic of the simple search Edit
Beware the search is done only on tickets with ActiveStatus, unless a keyword contains a status, for example "resolved" or "open stalled".
You can use id (any number) to jump right to the ticket, just type 10 and hit enter to jump to ticket #10.
You can use email address to search by requestor.
You can use name of a privileged user to search by owner.
You can use queue name to search tickets in that queue, but only if queue name has no spaces.
You can prefix a word with "fulltext:" to search in content of tickets. Don't overuse this feature as full text search without indexing can be very slow.
Everything else is searched in subject of tickets.
Behaviour may be changed or extended with local customizations, consult with your system administrator if you are not sure.
Some examples with magic Edit
Search for open or stalled tickets owned by ruz (if a user with name ruz exists).
ruz open stalled
Search for new tickets in support queue (if a queue with name support exists).