How to move magento search bar from header to sidebar?

Edit "app/design/frontend/<your theme>/<your theme>/layout/catalogsearch.xml"

Fine the line:

<reference name="header">
    <block as="topSearch" name="top.search" template="catalogsearch/form.mini.phtml" type="core/template">
</block>
</reference>

Replace the reference name to left or right.

<reference name="left or right(your wish)">
    <block as="topSearch" name="top.search" template="catalogsearch/form.mini.phtml" type="core/template">
</block>
</reference>