At a wedding, an official person must perform the ceremony. In most states, a justice of the peace can perform a wedding, as can a licensed or ordained clergy member. The differences between the clergy qualifications vary by state.
Ordination is a status given by a church. A minister can become ordained in a variety of ways, from sending in an online registration form to attending seminary for a few years. An ordained minister can perform all of the duties of a church leader, including leading services, preaching and performing baptisms.
In some states, a minister must be licensed to perform a marriage ceremony legally. In this case, the minister must apply for his license with the county or state government. A license is only needed for weddings; anything else a minister performs does not require the use of a license, since nothing else is a legal ceremony.
Some states require that a clergy member be only ordained to perform a wedding; in this case, anyone with a certificate of ordination from any church can perform a wedding. In other states, that minister has to also have a license, which carries other requirements.
In some states, anyone with an ordination certificate can also get a license. Other states require the clergy member to be the actual head of a congregation before being licensed to perform weddings.
In California, according to usmarriagelaws.com, anyone can apply for the status to perform a wedding, but only for one day. This means even your Uncle Bob can apply to become a "Deputy Commissioner of Marriages" and then perform your wedding ceremony, but he cannot perform other marriage ceremonies on other days.