A proposition is a blanket term for any ballot measure to be voted on by the people – this can be an initiative or a referendum.
An initiative is a brand new law or constitutional amendment proposed and voted by the people. It is a law started by the people.
In California, there is a direct initiative process, where a petition that receives the required number of signatures automatically qualifies to be put on the ballot. In other states, there is an indirect initiative, where, after receiving the specific number of signatures, the legislature decides if the measure will go on the ballot or not.
For example, legalizing marijuana and gay marriage have been put on the ballot through the initiative process in several states.
A referendum is when the voters decide to approve or reject an existing law. There are two different ways this can occur: one, if there is already a law in place, the voters attempt to repeal the law; and two, if the legislature decides to not vote on the measure and places it on the ballot for voter consideration.
Referendums typically happen when changes are made to state constitutions or taxes.