Update: October 17, 2014
U.S. District Judge John Sedwick has barred Arizona from enforcing a 1996 state law and a 2008 voter-approved constitutional amendment that outlawed gay marriage. He ordered the state to "permanently cease" its ban on gay marriage. Arizona's Attorney General has announced that he will not appeal that decision. He issued a letter to county clerks stating, "Effective immediately, the clerks of Arizona county superior courts cannot deny a marriage license to any otherwise eligible licensees on the grounds that the license permits a marriage between persons of the same sex." Same-sex couples in Arizona have already begun applying for marriage licenses.
Update: October 8, 2014
The U.S. Court of Appeals for the 9th Circuit, which has jurisdiction over Arizona, has declared marriage restrictions unconstitutional, citing that ruled the Idaho and Nevada bans violate couples' rights to equal protection under the 14th Amendment. The ruling is being appealed before the 9th Circuit panel. If the decision is upheld, Arizona same-sex couples could be marrying by the end of the year.
Last updated: Feb 2014
The short answer is...no. Arizona does not permit same-sex marriages. Only a union of one man and one woman is recognized as a marriage here.
Here is a bit of history about some of the ballot proposals in recent years relative to same-sex marriages.
2006: Protect Marriage Arizona
2008: Marriage Protection Amendment
Proposition 102 would amend the Arizona constitution by adding the following wording to the existing section on marriage: only a union of one man and one woman shall be valid or recognized as a marriage in this state. Proposition 102 passed with 56% of the voters voting yes.
An Arizona City Recognizes Civil Unions
In June 2013 the City of Bisbee in southern Arizona (population approx. 6,000) became the first community in the state to offer civil unions, with the City Council approving by a vote of 5-2. When originally proposed, there was concern on the part of the Arizona Attorney General's Office that there would be conflict with Arizona state laws, but a few revisions put those concerns to rest such that, at least within the city boundaries, any two adults, regardless of their gender or sexual orientation, may form contractual agreements and designate each other as agents.
There is a $75 fee in Bisbee to obtain a Civil Union Certificate.
The Future of Same-Sex Marriage in Arizona
was collecting signatures to try to get the Equal Marriage Amendment on the ballot in 2014, but those efforts were suspended in 2013 due to lack of funding. Other groups have hinted that it has a better chance if it appears on the 2016 ballot, when voter turnout is expected to be much higher than in 2014.