I have a contribution page set up for a flood recovery fund and some people want to give but not be recognized for their gift.
I am assuming you are publishing donor names somewhere and the donor does not want their name published. If this is correct, there are a few ways to handle this, below are a couple ways we do.
One way is to use a custom contribution field. It can be a yes/no or a check box with the question: Do you want your contribution to be anonymous? When running a report of names to publicize, you can filter out those that said yes.
Outside of donation pages, if we get an envelope of cash dropped off to our door with no contact information, we decided to create an organization called Anonymous Donations, where we log their contributions.
Also welcome to stack Aaron - just a reminder - try to include what version of Civi and your CMS (wordpress, drupal, etc) you are using. Sometimes there are answers specific to your setup.