what does it mean when they say to put a letter on letterhead?

You need to get letter paper from that organization that you are collecting donations for so the donors know your plea is legit. The letterhead paper will have the organizations logo (identifying trademark sign--like the "Coke" script--no one else can use it without paying for that use), address and usually the organization's tax I.D. number (for tax purposes). By using the letterhead, you are saying you have been authorized by the organization to collect money for them. If the money never gets to the organization, the donor can report your to the police for fraud and several other things.

Letterhead is paper that has a heading on the top of the paper (hence letterhead). I didn't know what it was until I worked for a company (a law firm) and the firm's name was at the top. It also feels like resume paper; it's the company's "special" paper they use when they send letters.