It is a social welfare organization that is not primarily involved in politics. They have two features attractive to interest groups that wish to be involved in politics from the outside - that is - not directly tied into a campaign.
1 - They are tax exempt.
2 - They don't have to disclose their donors.
This makes the designation attractive to many political groups that would like to register as one, but this is a call made by the IRS, which leads to the obvious controversy.
For detail:
- Open Secrets: Outside Spending: Frequently Asked Questions About 501(c)(4) Groups.
- Washington Post: What is a 501(c)(4), anyway?