In the United States, a charitable organization is an organization that is organized and operated for purposes that are beneficial to the public interest, however a distinction is made between types of charitable organizations.
Every U.S. and foreign charity that qualifies as tax-exempt under Section 501(c)(3) of the Internal Revenue Code is considered a "private foundation" unless it demonstrates to the IRS that it falls into another category. In a general sense, any organization that is not a private foundation (i.e. it qualifies as something else) is usually a public charity as described in Section 509(a) of the Internal Revenue Code.
If you are looking to set up a charity that is an eligible 501(c)(3), then Bit-Pay will handle the payments through their system at no charge:
If you are just looking for a way to let others contribute for a particular cause and not for a formally registered charity you can handle that in whatever manner you wish, including an online wallet such as My Wallet from http://BlockChain.info/wallet
If you wanted a website for that you could use something like Weebly, which is free: http://www.weebly.com
Of course, fundraising is very challenging. Here's the "accepts donations" list: