How is a system of axioms different from a system of beliefs? Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?

There are similarities about how people obtain beliefs on different matters. However it is hardly a blind faith.
There are rules that guide mathematicians in choosing axioms. There has always been discussions about whether an axiom is really true or not. For example, not long ago, mathematicians were discussing whether the axiom of choice is reasonable or not. The unexpected consequences of the axiom like the well ordering principle caused many to think it is not true.
Same applies to axioms that are discussed today among set theorists. Set theoretical statements which are independent of the ZFC. There are various views regarding these but they are not based on blind belief. One nice paper to have a look at is Saharon Shelah's Logical Dreams. (This is only one of the views regarding which axioms we should adopt for mathematics, another interesting point of views is the one held by Godel which can be found in his collected works.)
I think a major reason for accepting the consistency of mathematical systems like ZFC is that this statement is refutable (to refute the statement one just needs to come up with a proof of contradiction in ZFC) but no such proof has been found. In a sense, it can be considered to be similar to physics: as long as the theory is describing what we see correctly and doesn't lead to strange things mathematicians will continue to use it. If at some point we notice that it is not so (this happened in the last century in naive Cantorian set theory, see Russell's Paradox) we will fix the axioms to solve those issues.
There has been several discussion on the FOM mailing list that you can read if you are interested.
In short, adoption of axioms for mathematics is not based on "blind faith". 
