I have heard that there is a restriction from political involvement for 501(c)(3) organizations. What, if anything, does this mean?
A 501(c)(3) organization may not intervene in an election by supporting or opposing any candidate for public office. A public charity may engage in lobbying so long as it is not a substantial part of its activities. See the set of Ready Reference Pages on Lobbying and Electioneering for more details.
The so-called Johnson Amendment that prohibits charities, particularly churches, from participating in political activity may be repealed or restricted is occasionally considered by Congress. See Ready Reference Page: Commentary: Keep Charities Out of Politics.