Vim case insensitive search

A per search way is this:
/pattern\c

future searches case insensitive
:set ci or :set caseinsensitive

Smart case-insensitive, incremental search using vim

Smart case-insensitive

:set ignorecase (and hit return)
:set smartcase (and hit return)

With both ignorecase and smartcase turned on, a search is case-insensitive if the search string is in ALL lower case. For example, searching for apple will find Apple and APPLE.

Incremental search

:set incsearch (and hit return)

incsearch stands for incremental search. It means that you will see what vim matches as you type in each letter of your search string (without having to hit return before search is even attempted).