In a nutshell...
(1) A company cannot sell securities (like stock) unless the stock has either been registered (i.e. the company has had its IPO) or there's an exemption from the registration requirement
(2) The most common exemption is something called Regulation D.
(3) Regulation D allows sales to non-accredited investors as long as those investors get a fairly substantial set of information about the company.
(4) Regulation D allows sales to accredited investors, in which case the company has to disclose a lot less information.
(5) #4 above is why people sometimes think that only accredited investors can invest in private stock offerings.
(6) There are other exemptions which may apply. For example, you'll hear references to Section 4(2) of the Securities Act of 1933, which contains an exemption for sales made "not involving any public offering." This exception is often used for early-stage friends-and-family offerings.