A membership interest represents an investor's (called a "member") ownership stake in an LLC. A person who holds a membership interest has a profit and voting interest in the LLC (although these may be amended by contract). Ownership in an LLC can be expressed by percentage ownership interest or membership units.
An LLC is different from a corporation in that it is able to distribute its membership interests in any way it wishes, regardless of the amount of capital contribution a member makes to the company. In addition, an LLC can have different classes of membership interests, which enables it to allocate profits and voting rights in a special manner.
A membership interest is frequently represented in the admission of the investor's name to the LLC to a ledger, this means no certificated security instruments may be issued to membership interest holders aka members.
If an LLC is profitable, the Manager (equivalent to CEO) may make distributions to member's on a pro-rated or other basis. Therefore like holding stock entitled to dividends, holding membership interests may entitle a holder to distributions.
If an LLC is sold or merges, the members will receive cash or other assets as consideration for their ownership right.
LLC's generally do not go public, but may be converted to a C-corp in anticipation of a public offering, this would make membership interest holders stock holders.