In the United States, marriage means an agreement in which two people promise to live together for the rest of their lives. It is ALWAYS a civil contract--papers are officially filed and it has an official legal status. It is OFTEN a religious ceremony or sacrament. It is often both things at once--when some friends of mine were married, the priest said "And by the authority granted in me by the Episcopal Diocese and by the Commonwealth of Massachusetts, I pronounce you husband and wife."
The typical custom is a "church wedding," a combined religious and civil ceremony. It is however quite common to be married "in judge's chambers" or "by a justice of the peace." For example, one of my children had a church wedding, and another was married in a non-religious ceremony by a justice of the peace.
Many of the customs and ceremonies are similar whether the marriage is civil or religious. There is usually an "exchange of vows" or promises. The exact language of the vows varies, and some couples write their own, but a very common tradition is to use some variation of the language in the Anglican (Church of England) book of common prayer, often including language like this: "For richer, for poorer, for better and for worse, in sickness and in health forsaking all others, 'til death do us part."
There is currently controversy in the United States as to whether marriage is always the union of a man and a woman, or whether there can be marriage between two people of the same sex. Within the last ten years or so, about a dozen states have begun to officially recognize marriages between two people of the same sex. That is marriage in the civil, legal sense. Whether two people of the same sex can have a religious wedding depends on the particular religious denomination.