This ceremony changes your marital status from single to married.
Mexico's Matrimonio Civil and resultant Acta de Matrimonio (your proof-of-marriage certificate) is recognized by the United States, Canada, France, England, Scotland, Ireland, Germany and most other countries.

The Matrimonio Civil is performed by the Registro Civil via their justices.
The justices perform these marriages in the offices of the Registro Civil or, for an additional fee, just about anywhere you specify.
The Justices allow the saying of personal vows.

Required documentation for bride and groom: (Civil Wedding)