The previous two answers show you how to disable windows search altogether. This also causes search boxes in various places to disappear, most notably the search box on the bottom of the Start menu and the search box at the top right of file explorers.
Personally, I like the search box in various places, I just don't want an indexing process to be running all the time. The main advantages of indexing are that it leads to faster searches, and to in-document searches and email searches. If you need neither of these, but want to keep Search in general, proceed as follows:
Do not turn off windows search, or switch it back on via OptionalFeatures.exe. (reboot if necessary)
Go to the services control (either via Computer, right click, Manage; or by typing services.msc in the search box at the Start menu).
Find the Windows Search service, right click, Properties and set it to disabled at start up.
Reboot (not strictly necessary, but you won't be able to see whether it worked until you reboot).
Note that documents created after you do this will not be found by the search boxes. Windows search is based on indexes only and does not do the equivalent of a unix find anymore (as it did in the XP days).
You can occasionally switch on the Windows Search service and leave your computer on overnight to have it update the indexes.
A very good in-depth discussion of windows search can be found over at wikipedia.