This is truly unique with each state. Some states do not require you to report this information, although most have some reporting requirements. So in general, you can report changes to your Board by...
Filing Your Annual Report
Most states have an annual reporting requirement, whereby each year you'll pay a fee and report the most current information about your corporation, such as the company's address, the registered agent, and the directors. Every state is different, in what is reported, when a report is due, how it is filed (many are online), and the annual fee.
So if your directors have changed, and your state has an annual reporting requirement, you'd wait to file your annual report, reporting what is the current situation at the time of the report. But what if the annual report filing date is several months away and you need to update your director information publicly before this?
This might actually happen because of a bank loan, or as part of an agreement to remove a director, or for some other reason. If the annual report is not due for a while, you could always file an Amended Annual Report. This usually involves filing the same report, as previous, but with different information. You'd also need to pay the state's fee, though in some cases (i.e., California), there's no cost or it is reduced to make a change.
So what if director information is not part of the annual report? Then your only option is...
Filing an Amendment to Articles of Incorporation
Assuming the initial directors were named in the articles of incorporation, you could file an amendment to the articles, changing the name(s) of the director(s). Again, this is a public reporting of a change that has already taken place. And with most states, this is NOT required because the named director(s) is/are "Initial Director(s)." These are the corporation's 1st director(s) who are there until the first shareholder meeting, where a board is elected. So the state recognizes that the board of directors may change from time to time, and they certainly don't expect you to file an amendment each and every time you make a change.
Finally, as a reminder, filing an amendment does not change the director(s) (that's done at the annual shareholders meeting, special shareholders meeting and/or as described in the corporation's bylaws). This filing only updates the public record.