Your letter should say you are requesting information under the Federal Freedom of Information Act, or the applicable state statute. It should explain what information you are seeking. The narrower the request the more likely you are to get a favorable response and the sooner you'll probably get it.
If you are using the information for news, research or educational purposes, tell the agency that because you may be charged reduced fees or you may convince the agency to waive search and copying fees. I'll talk more about them later.
The agency may respond by providing the information, or may claim the information is exempt, or it may say "the cost for processing your request is estimated to be . . ." and that it will locate and send what it has if you first cough up the cash. Obviously, the first response is the one you most want to hear. You can sometimes facilitate things if you say in your request letter that the you'll pay up to a certain amount for the information, indicating that the agency should go ahead if the cost will not be greater.
There are nine exemptions to the federal FOI Act covering national security data, internal agency policies, trade secrets, predecisional inter- and intra-agency memoranda, personal privacy, law enforcement records, certain financial institution data, and data on oil and gas leases. There is also a catch-all exemption stating that if another federal law makes records secret you cannot get them under the FOI Act. A classic example is tax returns. The IRS Code states that they may not be disclosed and the FOI Act cannot be used to get access to them.
Although I have talked mainly about the federal FOI Act, many state laws are very similar. The exemptions may differ significantly. Most state laws, like the federal FOI Act, only apply to executive branch agencies. However, some cover non-governmental entities that expend public funds. A few cover the legislative branch as well. This has been a very general discussion.