If you refuse to serve, and if the person so refused can establish that s/he is a member of a 'protected class' (black, Hispanic, gay, disabled, some others) and if said person can also establish that the reason why you refused to serve is because of his/her membership of the protected class, you're in deep, deep, DEEP trouble and sinking fast. Even if it can't be proven that the reason why you refused is because the person is in the class, you've got a problem. Unless you can establish some clear, precise, and compelling reason which does NOT have anything to do with your problems with the class.
This means that if you are in the business of making wedding cakes and a gay couple orders a cake, you're gonna have to make that cake unless you want to go to court and establish why not. And you'd better be able to establish that you _couldn't_ make the cake for reasons entirely unrelated to the gay couple's gayness, or it's game over. If you have a waiting list three months long and couldn't make a cake for the couple anyway, you're okay. If you have workers standing idle and an unused oven, you're in trouble.
Now, if the persons who want the cake are _not_ members of a protected class, then you can refuse to make the cake all you want. At the current time, gay couples _in Arizona_ are not members of a protected class. This may change. This means that this idiotic law is completely unnecessary; Arizona businesses could, and can, simply post a note that they refuse to serve gay couples, and that's that. There is one gentleman who has posted a large sign in his business that he reserves the right to refuse to serve members of the Arizona legislature. http://www.cnn.com/2014/02/26/opinion/rocco-pizzeria-arizona/index.html?hpt=hp_t1 As members of the legislature are not members of a protected class, he is free to do so. And he has.
This 'law' is nothing more than religious-themed imbecilic grandstanding, and _will_ bite the Arizona legislature on the ass. And it won't last long before it gets challenged and gets to the US Supremes and they rule that it's unconstitutional... and place gays in the 'protected' class. I suspect that this is not something that the religious nuts want.