Like state sales tax issues, my limited experience indicates that this varies widely from state to state, and it's also called different things. Some states also call it a franchise tax.
Here in Virginia, they do call it a corporate income tax. But nonprofits only have to file if they file any unrelated business income on the federal level. I've often wondered how they enforce that--if they have some reporting thing with the IRS, or what--because if you didn't incorporate, Virginia might not even know you exist. There is no requirement to file anything in Virginia to be tax-exempt (for state income tax--sales tax is another story).
Here's some info on other areas that I know about, although this information might be a little stale at this point--but just to give you an idea of the differences.
In North Carolina, you actually have to get a letter from the state saying that you are exempt from state income tax. I think that they give it to you when you incorporate in the state, and there's not a separate application, but I'm not sure. Some types of nonprofits (but not PTOs/PTAs) then need to file a copy of their 990 with the state; they don't have a special state form. As with Virginia, though, if you have unrelated business income at the Federal level, you need to file at the state level.
In Massachusetts, you would just send in a copy of your Federal exemption letter, and you're set in the state. But all nonprofits also have to file an "annual report" with the Corporations Division--basic info on officers and stuff like that. I believe you also have to file a return if you have unrelated business income (that's probably a general rule throughout).
In Washington DC, you used to have to send a copy of your Federal 990 in every year, and it went to the Corporations Commission or something like that--not the Department of Revenue. I'm pretty sure that's changed now, though, so that you have to actually fill out an application for exemption with the Revenue office. You can file for exemption prior to receiving the Federal exemption, but I think they make a "preliminary" determination, subject to whatever the IRS ends up saying.
I've never seen a website like Pals showed for the state sales tax filings that outlines the income tax requirements, but even if I did, I don't know that I would trust it. (Especially since the info on Virginia sales tax exemption is incorrect on that website she references!) It's always better to go directly to the taxing jurisdiction.