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In regular expressions, the carat symbol means "at beginning of line"; so, in order for a match to happen with your given example above, the word "processor" must be found at the beginning of a line.
That's what it does in the example you cite (with grep rather than as anything specific to bash) but it's an overloaded symbol - it can also negate matches in [...] expressions or exponentiate (bc) and several other things in other contexts. Just saying that, if you see ^ in a shell script, you have to know the various contexts.
Caret is what I most often see it referred to as, and I cannot think of another name for it, although there are probably some. Here's the wiki page: http://en.wikipedia.org/wiki/Caret
Note that I had been spelling it incorrectly, but have now corrected it: Carat --> Caret
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