In the American university system, how relevant to the graduate admission committee is the institution from which a given candidate obtained his/her bachelor's degree?
The general answer is
Very relevant, but considered in the context of the rest of the application.
It seems hard to say more in the absence of a more focused question. Before you do that, you might try to search the site: I can't say that all possible versions of this kind of question have already been asked and answered here....but certainly many of them have.