In the US, our legislative branch of the US government is called the Congress. It is has two parts: the House of Representatives and the Senate.
The House is made up of representatives elected from "districts." Every state gets a certain number of districts based on that state's population in relation to the overall population of the US. So the more population a state has, the more representatives it has. A state is divided up into geographical areas called districts, and each district has the same number of people in it. Those people elect their representatives. There are 435 districts in the US, so there are 435 representatives in the House.
Each state gets to elect 2 senators to the Senate. So there are 100 senators.
For something to become a law, it must be separately voted on by the House and the Senate. If it passes both, then it goes to the President. If the President approved it, it becomes a law. If he doesn't approve it, then it doesn't become a law.
Now, in the US, the Constitution says that the US government can make laws about certain things and the states can make laws about other things. Generally, the U.S. Government can't make laws about the things that states are allowed to make laws about. (In practice, it's a little more complicated than that, but that's the general idea.)
One of the things that the states have the right to make laws about is marriage.
However, the Constitution has something called the 14th Amendment. It says that state laws must provide "equal protection of the laws" to its people, even if those laws are the kind that states have the right to make.
So, what is happening with the marriage issue in the US is that some states have passed laws which forbid same-sex marriages. Same-sex couples who have been effected by those laws have asked U.S. judges to find those laws invalid because they violate the Equal Protection Clause. Most courts have done so.
The President can't make any laws governing marriage. But the President runs what's called the Department of Justice. That's the government's lawyers that handle lawsuits and criminal prosecution. That department sometimes writes what are called "amicus briefs," which are its opinion on how a court should decide a case. They are used when the government has an interest in a case but is not a party in the case. The courts read these opinions but do not have to follow them.
In June, the Supreme Court (the highest court) will be deciding once and for all whether state laws against same-sex marriage violate Equal Protection. Many lower courts have said that it is a violation, but some other lower courts have disagreed. (There are many US courts spread throughout the country, and each handles cases from a different geographical region and different states.) So in June the Supreme Court will decide which group of lower courts is correct. And the President has had the Department of Justice submit an amicus brief to the Supreme Court which says that its opinion is that laws against same-sex marriage violate the Equal Protection Clause.