There is a lot of information out there about SB1062, so let's try to break it down.

How this came to be: There's a group called the Center for Arizona Policy and they were scared of a case in New Mexico where a lesbian couple sued a photographer for not photographing their wedding (they were able to find a cheaper photographer in time to shoot their wedding, but still sued for discrimination and won the lawsuit). So this group proposed a bill and got Sen. Steve Yarbrough to sponsor SB 1062 in the Senate and Rep. Eddie Farnsworth to sponsor another version in the House.

Legal issues: The LGBT community is not a protected class in Arizona or the United States, but they are a protected class in some cities, like Phoenix and Tucson. So technically, if you're gay you could be denied service from a gas station in Casa Grande right now, even without this law because you're not protected. However, with this bill, state law trumps city law which could cause problems in Tucson and Phoenix.

How the Democrats see this bill: This is a discrimination bill. This bill was designed for the sole purpose of discriminating against the LGBT community. Even though there is no law protecting the LGBT community now, this will provide a safeguard years from now when there is a law protecting the LGBT community (Republicans can see the writing on the wall). Even if this bill doesn't defend people who are discriminating against gay people, it will give them a "legal" defense that can only be overturned in court. That means theoretically a businessman can turn a gay couple away and just take his chances on whether or not he will be taken to court. This will hurt business in Arizona. Just look at the companies that are calling for the Governor to veto the bill. If she signs this into law the economy will be significantly damaged because companies won’t want to move to Arizona’s hostile culture.

How the Republicans see it: This bill is being blown out of proportion. The LGBT community isn't protected, so this changes nothing for them. All it does is protect people's religious beliefs. People who are religious shouldn't be forced to violate their religion because the government says so. This bill also tightens up the language on the existing Religious Freedom Restoration Act from 1999 (similar to the federal one from 1993) and makes it so that there has to be a substantial burden on a person's religion in order for a business to turn them down. Their goal is so that it protects churches that don't want to marry gay couples or doctors that don't want to perform abortions. It's not intended to protect bigoted business owners that don't want gay people's money. The Democrat’s claims that people will be discriminated based on religion, race and sex are unfounded. All three of those classes are protected by federal law, which trumps state law.

Where it stands: The bill passed through the Senate last Wednesday and the House last Thursday. It’s now on the desk of the Governor waiting for her response.

Brewer’s Options:
Veto the bill—this is the option that’s getting the most play right now. There have been protests in Tucson and Phoenix calling for the governor to veto the bill because of the effect that it will have on the LGBT community. Even some GOP, including Sen. Steve Pierce, Sen. Bob Worsley and Sen. Adam Driggs (all of whom voted for the bill in the Senate) are calling for the bill to be vetoed. It’s gotten some national play as well, with both Arizona Senators and even a Delaware Governor weighing in favor of the veto.
Sign the bill into law—While the most vocal of opponents are calling for a veto, there are still many supporters of the bill. Rep. John Kavanagh went on CNN in defense of the bill last Friday and Rep. David Livingston continues to tweet out links to articles where LGBT members sued people.
Let the bill become law without putting her name on it—In Arizona, if the Governor doesn’t sign the bill within 5 days of it being on her desk, the bill becomes law. If governor Brewer chooses this option, she can disassociate herself from the bill while still allowing it to go into law. It’s hard to imagine this happening because of how vocal people are being right now so she can’t simply ignore it.

Why it will most likely be vetoed: This will probably be vetoed because the business community cares. Governor Brewer is a deeply religious, conservative lady, so she most likely supports this bill. However, this is her last year in office and she's working on her legacy. She "brought" Arizona out of the recession and her whole theme this year is bringing business to Arizona. If she signs this bill into law, she ruins that pro business reputation and she becomes that lady who signed those two bigoted bills into law (SB1070 and SB1062). She doesn't want that.