I just went to my K-1 fiancé visa interview at the U.S. Embassy, thinking I had all of my paperwork together and was ready, and the embassy officer denied me the visa! It wasn't that there was any big thing wrong. But the whole time, he kept asking me questions about my relationship with my fiancé in the U.S., and I just felt like he didn't like my answers. Then at the end he said I had not convinced him that this was a real relationship.
How could this happen? We were planning such a beautiful wedding! Can I appeal, or somehow get a second chance at the interview?
You are facing the same situation as many applicants have, and it might not be entirely your fault. If you are from a country where many people have filed fraudulent fiancé visa petitions in the past, the U.S. State Department official who met with you might simply have taken a harder look at your application than is usual, and as you say, remained unconvinced.
Unfortunately, you cannot file an appeal (for this or any other type of nonimmigrant visa decision). There is just no procedure for this.
One possibility is that the U.S. embassy that denied your visa could send your fiancé's I-129F petition back to U.S. Citizenship and Immigration Services (USCIS). USCIS then would send your fiancé a "Notice of Intent to Revoke" the petition and afford an opportunity to overcome the basis for the visa denial overseas. If successful in overcoming the basis of the denial, there eventually will be a new visa interview.
The timing of this can be unpredictable. It might be several months or up to a year or longer before hearing from USCIS and then several more months for the new interview.
Starting from the beginning with a new I-129F fiancé petition is a possibility. You would need to redo all the same paperwork and pay a new set of fees. This would, of course, lead to another visa interview, probably at the same U.S. embassy as before.
And no matter what embassy or consulate you are scheduled to go to, a record of the denial will remain in your file, possibly leading to another denial. So this strategy might or might not work.
For some applicants, the best thing to do is to get married overseas and apply for an immigrant visa. An actual marriage shows a higher level of commitment by both the would-be immigrant and the U.S. citizen, and is harder for U.S. State Department officials to dismiss when making the visa decision. Besides, more time will have passed by then, making your case look more convincing. (See the articles in the Marriage-Based Visas and Green Cards section of Nolo's website for more on this.)
If you had your heart set on a wedding in the U.S., there is nothing to stop you from having a big ceremony there even after you have already legally married in your home country.
You might wish to consult with a U.S. immigration attorney for a full analysis of what went wrong with your case and how to correct it and succeed in your request for a U.S. visa. The attorney will know the approach taken by the officials at the U.S. embassy you visited, and may be able to spot problems in your application of which you were unaware.