"Plebiscites" are "political issue" petitions that gather a bunch of signatures, but do not change any law (much like a "poll," but in the form of a petition page used to gauge support for an issue).
"Referenda" (the plural of the singular "referendum") are petitions that place one of a legislative body's recently-enacted laws on the ballot, for the purpose of altering or abolishing it. (The legislative body could be a "city council" for a municipal referendum, or a "state legislature," for a state-wide referendum.)
I've been a libertarian ballot access petitioner in the USA for 14 years. In that time, I've circulated petitions to place philosophically libertarian candidates on the ballot(as Independents, Libertarians, Constitution Party, Democrats, and Republicans). I've also worked to place pro-liberty issues on ballots as "ballot measures," which will make a change to the law if enough people vote for them.
I don't track the minutiae as well as some petitioners I know, but you'll be better-served by my response than most of the others here. Especially if you want to understand what is happening in the USA. I also highly recommend "Citizens In Charge" a pro-initiative and referenda website that helps place citizen-sponsored initiatives on ballots in the 23 States that allow this process. (Illinois would make it 24, but the initiative process in Illinois is false, since the legislature and courts can throw the initiative out, since they're "non-binding." This just happened with the "Term Limits" petition in Illinois the courts did their friends in the legislature a favor and ruled it "unconstitutional" even though the voters of Illinois voted overwhelmingly in favor of it. The Constitutional Amendment process in Illinois is also limited to a few narrow subjects by the IL Constitution. These narrow limitations regularly violate both individual rights and the will of the people, resulting in a useless "CA" process in IL.) For a full reference map, please visit: http://citizensincharge.org/states
Initiatives, Referenda, Constitutional Amendments, and Plebiscites are all means of addressing political "issues." The first three(I, R, & CA) are forms of "ballot measure" which access election ballots and directly make changes to laws(either adding new laws, or abolishing/repealing old ones). The last one(plebiscites) simply gauges public opinion on a political matter, agitates the general public around an issue for purposes of show, or organizes people into a group for possible fundraising or other political action (often times, a plebiscite precedes a "ballot measure").
1) Initiative: A citizen-sponsored "initiative" places a potential law (usually dealing with a single "issue") on a ballot. That "initiative" is then voted on by the people in the first election that comes after it gathered enough signatures to be placed on the ballot. If it gets enough signatures (50% of the vote in most places; 60% in Florida, etc.), then the "initiative" becomes a law. The legislature can also place an issue on the ballot (but since they can also make laws directly, this typically happens only when they want to avoid the blame for passing a law).
2) Referenda: A referenda generally happens when the legislature passes a law, and people object to the law, so they say "We're going to gather enough signatures to 'refer' this to the voters, before it goes into effect." That's exactly what it does: it repeals a law before the law goes into effect. (If the law has already gone into effect, and the referenda passes, repealing the law, the punishments or other effects of the law are reversed, nullified, or waived.) Generally, there is a time period before laws go into effect which allows for referenda, should a law be unpopular enough. Sometimes, legislatures will refer their own laws to a vote, because this allows them to test how aware their constituency is, or how corrupted they are. For example: Most legislature-sponsored referenda will increase taxes, or create some government regulation. If the voters are dumb enough to vote for it, the legislature can simply claim innocence: "I just put it in front of the voters, and the voters must have wanted it, because they voted it into effect." In Illinois, teachers unions often successfully try to guilt stupid parents into voting for referenda that will increase taxation (rather than spending existing tax money more intelligently).
3) Constitutional Amendment: This is the same as an initiative, but it changes a State's Constitution. Because the State Constitution is the highest law of the State, it is usually significantly more difficult to place a constitutional amendment on the ballot, and the state has a harder time claiming "That's not really what the voters wanted," if they attempt to circumvent the will of the voters.
4) Plebiscite: Plebiscites simply gauge public opinion on a subject. People are asked to sign a petition expressing a certain viewpoint, and these signatures are then able to be used to: threaten the legislature with an organized attempt to access the ballot if they do not take action, gather names for a mailing list. (Marijuana Policy Project has done plebiscites that asked people if they wanted to legalize marijuana, for example, while also asking those who wanted to legalize marijuana to give them their emails, so they could "stay in touch" and "inform the signers about possible changes to the law in their state.") A plebiscite generally does very little to change a law, or organize people in favor or against a law.
Any of the above, except a plebiscite, can abolish or create a law, or change a portion of a law.
A referenda deals with a specific law the legislature has recently passed, prior to its going into effect.
Initiatives create a "statute law" (of which there are thousands on the books, so many, in fact, that they cannot all be followed). States often claim they cannot enforce a statute law the way it was written, and often states prevail in essentially "ignoring the will of the voters" or "declaring a citizen-sponsored statute law unconstitutional."
In order to sponsor the referral of an existing law to a vote(referenda), sponsor a statute law(initiative), or sponsor a constitutional amendment(a change to the State Constitution) one must generally be an elector (a voter is an elector who actually votes in the election in question) in the state in which the ballot measure is being proposed. Sometimes, people use "initiative" to refer to any of these three things, even when they mean "constitutional amendment" or "referendum"(referendum is singular of "referenda"), and should have said "ballot measure."
Generally, the people who run plebiscites tend to do so for one of three reasons:
1) They lack the money and organization necessary to place a ballot measure on the ballot.
2) They honestly don't know what the general public thinks about an issue, and wish to treat interaction with the public as a kind of "informal polling."
3) They wish to make a video showing members of the general public reacting a certain way to an issue as a form of "psychological warfare" against the legislature, or as a means of selling ad money on a video-sharing site, or as a means of promoting their organization. Mark Dice's famous "Man on the Street" videos are plebiscites that often crudely pretend to carry the force of law, or to be "a binding initiative."
Similarly, an initiative that is done for popularity purposes can essentially have no effect. An initiative was once done in the Pacific Northwest to label a popular conservative activist "a horse's ass." This initiative was non-binding, so it essentially acted as a plebiscite, designed to draw attention to that conservative activist in an unfavorable light.
Another reason for running an ineffectual initiative that will not actually change anything is to trick the voters into voting for a seemingly "less radical" or "more compromising" law than an actual initiative that is being proposed by people who wish to legitimately change the law. (For example, in California, the police unions will sometimes sponsor a "fake" initiative that allegedly "urges the [existing prohibitionist] legislature [that is currently too stupid and corrupt to legalize marijuana] to legalize marijuana" when a "real" initiative petition is being circulated to directly force the state to legalize marijuana upon passage by the voters. Some naive petition signers will then sign the petition being sponsored or "backed" by the police, and will then tell petition circulators of the real petition that "they already signed for that." This then makes it harder for the real petition to gather enough signatures to access the ballot.
Hopefully, this clears up the basics of how the "initiative and referenda" ("I & R") process works in the USA.
For more information, definitely check out "Citizens In Charge" ...they're the good guys who want citizens to be able to change laws. They are often painted as "conservative" but are really libertarians. (Their chairman actually went to prison for refusing to register for the draft under Ronald Reagan.)
There are other groups that try to eliminate citizen choice, and expand the power of the state, often by making it more difficult to place "ballot measures" on the ballot. These groups often have deceptive names that make it sound like they are pro-initiative or pro-freedom when they are the direct opposite. (For example, a group called "Ballot Initiative Strategy Center" tries to create laws that make it illegal to pay petition circulators per signature, knowing that initiative sponsors will then be unable to recruit talented people to explain their initiative and get voters to sign for it. They sometimes lie and claim this is to make sure petitioners are being "paid fairly." ...LOL. Petitioners lack continuous, year-round work, and must make money in brief spurts of activity. If they're paid $12/hour, they won't make enough to justify working on the initiative, and will leave to find work in states that pay per-signature, and the initiative won't qualify for the ballot. If they are paid even $1/per signature, a decent petitioner will get 20 or 30 per hour, and make $30/hour, or $300 per day for a 10 hour day.) BISC also supports de-facto outlawing minor party candidates, by sponsoring a switch to "top two" primary elections, as they successfully did in CA and WA. This means that only the "top two highest vote-getters" in a primary election are listed on the ballot in a general election: even if it's two people from the same party. Since this is the case in CA, there are now two "Democrats" on the ballot for U.S. Senate in CA. The "top-two primary" claims to protect voting rights, but it really eliminates them, consolidating power in the hands of the incumbent legislature.
A committee that sponsors a ballot measure is a cybernetic entity. As such, it is capable of dishonesty, as a means of self-preservation. Because of this capacity, noone should be surprised when "trickery" is used as a means of getting initiatives passed, fighting initiatives or causing them to fail, etc. All kinds of information will flow during a ballot measure campaign. Because ballot measures must be voted on by uninformed government-educated voters, much of this information will, by necessity, be "low-level." It will contain partial truths, because complete truths are often very uncomfortable. The opposition will ALWAYS use ad-hominem attacks. (This is because ad-hominem attacks are often effective. This is especially true since most people do not have a strong preference for government power to be limited or unlimited, but the sociopaths in government definitely want government power to be unlimited. Thus, intelligent sociopaths who have already successfully won power have a huge interest in misrepresenting issues that could reduce that power. Further, they are misrepresenting these issues to people who were dumb enough to elect them in the first place, so they already have trouble gauging political honesty.)
This tends to result in government power being expanded by initiatives as often as it is reduced by them. Some people who wish to limit government forget that the incumbent government virtually NEVER votes to limit government power, and that initiative and referenda are the only other means of making or repealing laws. (ie: 50% is better than 0%.)
I support expanding "the initiative process," and undoing the damage that has recently been done to it by placing restrictions on how signatures can be gathered (pay-per-signature bans; bans on out-of-state petitioners; requirements to wear a state-designed ID badge; etc.).