For religious purposes, those having a Jewish wedding record the Hebrew date rather than the secular date. To determine your Hebrew wedding date, I collect your secular wedding date and go to the HebCal Date Converter. It reliably provides the corresponding date.
Each of my standard texts starts with something like "On the __ day of the week, the ____ day of the month of _____ in the year 57__, which corresponds to ___________." The first part is the Hebrew date, the second is the secular date.
If your secular wedding date was Sunday, August 11, 2013, your Hebrew wedding date would be the 5th of Elul, 5773. When filled out it will read something like "on the first day of the week, the fifth day of the month of Elul in the year 5773, which corresponds to Sunday, August 11, 2013."
However, the Hebrew or Jewish calendar changes year to year, so next year, August 11, 2014 = 15th of Av, 5774. Your Hebrew anniversary, 5th of Elul, 5774, would actually correspond to Sunday, August 31, 2014.
So, yes, that means you have two anniversaries, what you do with them is up to you. Some people celebrate the secular anniversary with a nice dinner, etc., and then do something good for their communities on their Hebrew anniversary - volunteer together, make a donation to a good cause, or help an elderly relative. (If you manage this, I will be impressed. Most people are lucky to remember the secular date).
The other thing to know about Hebrew dates is that the Jewish calendar day begins at sundown, not at midnight as we are accustomed to with a secular calendar. So if you are getting married at 7 PM Saturday evening, your Hebrew date will be the "first day of the week" rather than the seventh day as you might expect.
Which leads to a frequently confusing issue - by far, Saturday evening is the most popular time for my couples. Some even begin their weddings at 4 PM on a Saturday afternoon. This presents a particular conundrum. You see, in Jewish tradition, weddings are not supposed to happen on Shabbat, the seventh day. You are supposed to wait until 3 stars are visible in the sky (or an hour past sunset in case you can't see stars).
A peculiar tradition has evolved amongst liberal Jews to accommodate both our secular need to have guests arrive and have dinner before 9 PM in the summer, and to keep some aspect of Jewish tradition alive in celebrations. The ketubah records the time of the wedding as happening after Shabbat ends. Most rabbis will require this. If you are a stickler for accuracy, please take it up with your officiant and let me know the decision.