Use of these terms can be confusing. Lawyers tend to identify organizations by the relevant section of the Internal Revenue Code, so that there's no ambiguity.
A 501(c)(3) organization is prohibited from endorsing political candidates, and is limited in its lobbying on issues. The organization is not taxed. Donations to the organization may be deducted by the donors on their individual income tax returns, because they are charitable contributions.
A 501(c)(4) organization is also a nonprofit and is also not taxed, but it is permitted to endorse political candidates. Contributions to the organization are not tax-deductible for the donors.
Most churches and other religious organizations in this country are 501(c)(3) entities, as are the Red Cross, various disease-related foundations, and a host of others. Some religious leaders have occasionally gotten carried away and skirted -- or, in my personal opinion, crossed -- the line against endorsements. The IRS has not made a widespread practice of cracking down on such instances.
Some of the organizations mentioned in this thread, such as the Sierra Club and NOW, are 501(c)(4) organizations. If you've been deducting your contributions to them, see your accountant about filing an amended return -- that's an improper deduction. Such organizations (also including the NRA, for example) are allowed to endorse political candidates.
It's a very common practice for a 501(c)(4) organization to set up an affiliated 501(c)(3) foundation, so that donors who want the deduction can make charitable contributions that will get c3 treatment while furthering the goals of the c4 organization. The foundation's funds can be used only for limited purposes, such as litigation. This restriction makes such money less valuable. (You'll sometimes see the terms "hard money" and "soft money" in this context. The hard money is what's harder to raise because the donor can't deduct it; it goes to the c4 organization and can be used with fewer restrictions. The soft money is what goes to the c3 foundation and can be used only for limited purposes. Sound management dictates that soft money be used for any project that qualifies for it, because this preserves the more flexible hard-money funds.)
Often, as in this thread, it's unclear whether "tax the churches" means "treat the churches as c4 rather than c3 entities" or "evict them from section 501 entirely." One point to bear in mind is that there are many, many c3 entities that aren't churches, and any change affecting churches will presumably affect all those other c3's as well.
And, yes, I'm aware of the 501(c)(5) and 501(c)(6) organizations. By omitting them I'm oversimplifying. Practically every sentence of this post is an oversimplification in some respect or other. It's a post, not the book-length treatment that would be required for complete accuracy.
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