We have generally two separate areas of jurisdiction to appeal to in case law: state and federal. Because federalism is so strong in the US (the political system in which sovereignty is constitutionally divided between a central governing authority, the federal government, and states) states can often exert their own authority to decide what law is best for their state. This is why from state to state laws can vary quite a bit in the way that they handle everything from capital punishment to drug law, and here to marriage and divorce for same-sex couples.
In the realm of family law, laws pertaining to marriage are created and governed by individual states. Texas statutes and the state constitution specifically prohibit same-sex marriages or civil unions. Citing those laws, Attorney General Greg Abbott successfully argued in a Beaumont case that two men could not get a divorce from a civil union in Vermont. So even though they were legally married in that state, and states always give "full faith and credit" to the marriages of opposite sex couples from state to state, the law is not so accommodating for same-sex couples.
When arguing against the recognition and dissolution of one specific out of state same-sex marriage, Attorney General Greg Abbot stated that "to prevent other states from imposing their values on this state, Texas voters overwhelmingly approved a Constitutional amendment specifically defining marriage as a union of one man and one woman, (their marriage) is not a marriage under Texas law, they are asking a Texas court to recognize - and dissolve - something that does not legally exist."
The federal government seems to be on the same path as Texas, leaving states where same-sex marriage is legal in the minority. In 1996, the United States Congress enacted the Defense of Marriage Act (DOMA), which had two major implications: it refused to recognize same-sex marriage on the federal level and it permitted individual states to refuse to recognize same-sex marriages regardless of their validity in other states. In Texas, this would leave only one option to same sex couples looking for marriage or divorce, they would have to appeal the decision all the way up to the Supreme Court asking that DOMA be found unconstitutional. No easy feat.
Another couple did not have to take it that far. Despite the protest of the Attorney General, in a separate case from a marriage authorized in Massachusetts, Judge Tena Callahan held that Texas' refusal to recognize the marriage was in violation of the "Equal Protection Clause" of the Fourteenth Amendment to the United States Constitution. This is the same amendment used to force states to comply with federal laws like Civil Rights Act. The decision was contrary to those of her peers, and could, at the least, be a step towards the recognition of out of state same-sex marriage. The case was appealed and if it the Texas Supreme Court accepts it, the clarity of the issue could be resolved, letting same-sex couples know where they stand. If it goes on to the Supreme Court, and is again accepted, it could affect federal law on same-sex marriage.