I know a person that is on the Board of Trustees for a company where I have applied for employment. My question is: is it ethical to contact that Board member about my application? Will I grease the wheels or clog the wheels?

I think if you know them well they might mention to the HR person to look at your resume or they might glance over it quickly and decide for themselves if you should be interviewed. But more likely than not (especially if they are just an aquaintance) they will just say "ill look into it and ask HR to get back to you" but actually do nothing. My guess is it won't change a thing.

Ask the Friend on the Board if you may use them as a reference on your resume.

Unless you are really good friends with the board member and are desperate for the job, I would advise against approaching him or her with any information. True, that so many jobs are obtained because someone knows someone. But, I still would not approach this person. Either the job is going to be a good "fit" between you and the company or it isn't.

If I were you, I'd like to think that my cover letter and resume carry enough weight on their own to get me an interview, and then how I show them that I am the best person for the job during the interview will land me the job. That shows self-confidence. Getting people to be on your side can be viewed by some to be a sign of weakness. Maybe, that you won't be able to do the job that is required without asking for help.

During your well-prepared for, and awesome interview because of your preparation, one of things that may be asked of you might be what you know about their company and how your having a job their would best serve their company. Since you will have done research on their company to more than adequately answer the question, you may be able to bring up the fact that you know someone who is connected to their company, and how much (I hope) that they enjoy doing what they do in connection with their company.

Another thing is, that interviews are a two-way thing if someone is well prepared for them. It is good to go into an interview with some questions of your own related to how you could best serve their company, what the goals of their company are, etc. But, not about what their company can do for you.