The Catholic Church believes marriage is forever. A divorced Catholic who wants a second wedding in the church needs a "declaration of nullity" from a church tribunal. The declaration says the previous marriage was invalid, so the second marriage is the first and only in God's eyes. This is unrelated to secular court decisions to grant an annulment or a divorce. It's internal Catholic business, so the church, not the government, keeps the annulment records.
How to Find Annulment Records
Each Catholic parish keeps records of significant events in its parishioners' spiritual lives: baptism, communion, marriage, death. Marriage records go into each parish's marriage registry. So do records showing the marriage has been annulled. It would be convenient if the church had a big database you could plug a name into, but that's not the case. Most records remain at the parish level.
If you need a copy of the declaration to prove that you can have a church wedding, contact the parish and ask for a copy. Churches are quite used to requests for sacramental records. Ask the parish what it requires to approve your request. Chicago's Holy Name Cathedral, for example, says you have to send in a letter with your name, the date of the annulment and a copy of your driver's license or similar ID. The parish might charge a fee.
Genealogy and History
If you're researching your genealogy or compiling a family history, you might be able to find records online. The United Kingdom has some parish records posted in its National Archives, for instance, but the records aren't complete, and you'd need to know the right parish.
You can also go to the parish in question and ask for help with your research. The Archdiocese of St. Paul and Minneapolis, for instance, accepts genealogical requests for records older than a century. More recent records are confidential, but your personal records or those of your children are always available on request.
Escaping Dead Ends
If you contact the parish where your ancestor lived and find nothing, don't despair. Do more research instead. Check neighboring parishes. Check other parishes where your ancestor's immediate family lived. Ask local historical societies for advice. If your ancestor married secretly somewhere else, perhaps the annulment record is in that other parish too. Keep going until you find the right parish, then make a request for the records you want.