Binary search ... is a search algorithm that finds the position of a target value within a sorted array. It compares the target value to the middle element of the array; if they are unequal, the half in which the target cannot lie is eliminated and the search continues on the remaining half until it is successful.
This video starts with searching a name in a long contact list. It gives good intuition because that is how we look things up in a printed phone book or dictionary. We simulate binary search ourselves, without knowing its called binary search.
It also shows you an implementation and walks through it. You only need to watch the first 4m:30s, after which he moves on to binary search trees.
Visualization of the binary search algorithm where 4 is the target value
The tutorial on TopCoder is amazing (you don't need to read the whole thing, the first two sections explain the algorithm completely).
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Source (for image, and block quotes): Wikipedia