This varies from State to State, but typical grounds for annulment include fraud, bigamy, insanity, and similar circumstances or conditions. Many States no longer use the term "annulment", but use "invalidation" in its place.
You would file for annulment the same way as any other lawsuit. A civil filing fee is paid through the clerk of the court, and a complaint or petition requesting the annulment (alleging the necessary grounds) is filed with the court and then served on the other party.
Does a divorce or anullment need to be filed in the same state the marriage was in?
No. You may file for an annullment from any State, regardless of where your marriage took place. Note that some States no longer use the term "annullment", but instead use "invalidate". The general procedure for obtaining an annullment or invalidation is substantially the same, as are the results. The marriage is dissolved and is presumed to be legally non-existant.