Organizations similar to 527s, with the following exception: unlike 527s, 501(c)s are not required to disclose their expenditures until after an election, they are not required to disclose their donors at all, and their primary purpose is not to affect elections. Generally, they only meet the first two criteria.
See "Bush's Secret Stash" by Nicholas Confessore, which discusses the differences between 527s and 501(c)s, why 501(c)s are far more insidious than 527s, why the government does nothing to stop them, and why the press doesn't care.
501(c)(3) organizations include many churches or other religiously-affliated groups. However, the status is also available and widely granted to several types of not-for-profit charitable and public service-oriented organizations, which can and does include grant-making foundations -- even those with a demonstrable ideological bent.
According to the Internal Revenue code: "all IRC section 501(c)(3) organizations, including churches and religious organizations, are absolutely prohibited from directly or indirectly participating in, or intervening in, any political campaign on behalf of (or in opposition to) any candidate for elective public office."
How 501(c)(3)s may avoid this prohibition
501(c)(3) organizations, however, are not entirely prohibited from lobbying, nor are their close cousins, the 501(h) and 501(c)(4)s, which were created to permit respectively greater expenditures on lobbying. This is a favorite "work around" for particularly active and ambitious politicians (see Newt Gingrich's Progress & Freedom Foundation and Abraham Lincoln Opporunity Fund; also Bob Dole's Better America Foundation, for example), who may channel tax-deductible donations from supporters into organizations dedicated to the promotion of general legislative and policy agendas that happen, oddly enough, to coincide with their own personal and political legislative and policy agendas. Some politicians have stretched this concept into very gray areas, as exemplified by Tom DeLay's Celebrations for Children, a 501(c)(3) organization whose primary goal appears to be to give large donors access to political allies of DeLay.
Although the IRS code prohibits 501(c)(3) organizations from spending a "substantial" portion of their resources on lobbying, many such organizations maintain parallel structures organized as 501(c)(4)s, which do the lobbying work while the 501(c)(3)s process the donations. 501(c)(3) organizations which do attempt directly to influence legislation and policy through the publications of studies and position papers typically include general disclaimers in their product, noting that, "Nothing herein is to be construed as necessarily reflecting the views of (this organization) or as an attempt to aid or hinder the passage of any bill before Congress."
In this way, 501(c)(3) organizations are able to indirectly influence elections, and even the prospects of individual campaigns, as when their advocacy targets the policy proposals of individual politicians. (By way of illustration: an economic analysis entitled, "Gephardt Tax Plan: Complex and Biased Against Savings and Growth," an actual (1995) publication of the Institute for Research on the Economics of Taxation.) Because the object of their advocacy is ostensibly the policy and not the politician, it is often deemed to pass muster under the IRC's seemingly ironclad prohibition against intervention in elections.