You are allowed to send a written argument (a "brief") to the Appeals Board to explain why you think the Judge's decision was wrong. If you decide to submit a written argument, you will also need to see the transcript of the hearing since the Appeals Board decides solely on the evidence presented at the hearing.
You must request a copy of the transcript as well as the opportunity to submit a brief at the time you request the appeal.
When you receive the Judge's decision, there will be two sections which are very important. The first section is the "Findings of Fact," and the second section is called "Conclusions" (based on those facts).
When you write your brief to the Appeals Board, you should refer to each Finding of Fact that you disagree with by the number the Judge assigned to it and you must briefly explain why you disagree with it. Also, you should refer to the appropriate page in the transcript where there is testimony on that fact which will show that the Judge made the wrong finding.
For example, if you were denied unemployment benefits because the Judge says you quit without good cause, the Judge may enter a Finding of Fact something like this:
"(2) She [you] did not discuss her problems with the employer before quitting the job."
If you did discuss the problems with your employer and you said this at the hearing, you should point that out and cite to the transcript. "Citing to the transcript" means pointing out to the Appeals Board the page number where you mentioned discussing the problems with your employer. You might write something like this:
"The Referee's Finding of Fact #2 is wrong. I did try to talk to my employer. See Tr. 12-13." (This means the evidence you gave at the hearing appears on pages 12 and 13 of the transcript.)
After you have explained why you disagree with the Judge's Findings of Fact, you should explain why you think the Judge's conclusions are wrong.