School choice has been a policy goal of many ed reform promoters for a while now, but for those not paying close attention, discussions of choice can be confusing because "choice" involves several different camps, camps that don't always necessarily get along. Each varies a bit depending on state laws, but here are the basics of how to tell them apart.
A charter school system begins with an authorizer--the organization that says, "Okay, you can open your charter school in that location." In some states, the authorizer is the local public school board (this is why some charter operators work to be elected to local school boards). In others (e.g. Michigan) other groups may be authorizers, even if they are hundreds of miles away from the charter school site. Authorizers are responsible for checking to see that the charter school is behaving itself, though again, as in Michigan, that can get murky when the authorizer benefits financially from the charter school's operation.
Once a charter is authorized and operating, students may shift into the charter from their public system, and some amount of money follows them into the charter, redirected by either the state or the local school board. The money may follow the student, but it is still managed by the state. Within the charter community, there is considerable disagreement about how much regulation charters should be subject to.
In a voucher system, the state hands you a voucher good for X amount of tuition dollars. You can now try to cash that voucher in at any school that will accept it (and will also accept your child). The school does not have to be "authorized" to exist, and you can try to cash your voucher at anything from Phillips Hoity Toity Academy to Bob's School of Stuff. Spoiler alert: your voucher isn't nearly enough money to get you into the high-end school.
While a charter system involves opening a whole bunch of new charter schools, vouchers usually feed a pre-existing batch of private schools--specifically, religious ones. For example, 97% of the schools receiving vouchers in Indiana were religiously-affiliated. And because these are private schools, they are free to reject your student for being the wrong religion or race or hair color (To clarify: It is technically illegal for them to do so, but plenty of them still find ways to let families know who belongs and who doesn't.)
But isn't it illegal to give tax dollars to religious institutions? Well, that's the beauty of vouchers-- the government didn't give money to the religious schools. It gave money to parents, and the parents gave it to the religious school. Kind of like when the fourteen year olds in your neighborhood gives their twenty-two year old neighbor money to buy them beer. Totally legal. Ish.
One other wrinkle with vouchers--depending on how the law is written, voucher money may go to students who never actually attended public school in the first place. When a law like this is implemented, public schools suddenly lose a bunch of money even though their student population reduces by None.
Education Savings Accounts are just one name for this approach to choice. Instead of a tuition voucher, you would have an educational account--perhaps an education debit card with X number of dollars on it. You could use this for private school tuition. Or if you wanted to home school, you could buy supplies. You could hire a coach to help your child do well on the SATs. Or, depending on how carefully the law is written, you could take an educational trip, or buy a PlayStation with some educational games.
Super-vouchers appeal to folks who envision an educational future in which students pick up a math course on line and a language course at a local academy and get a history credit by reading some god books and just generally a la carte their way to an education. But even some of my very conservative friends balk at the teeny tiny amount of oversight given to the spending of taxpayer dollars in these programs.
While all of these approaches come under the school choice umbrella, they do not necessarily play well together. Some charter supporters see vouchers as creating a really tight market to compete in; others are not comfortable with the tax-dollars-to-religious-school aspect. Some voucher supporters tolerate charters only as a stepping stone to get to full-on vouchers, because only vouchers will let tax dollars flow freely into religious schools (Betsy DeVos may well fit into this category). Voucher fans want full freedom from government oversight and worry about government regulations following the tax dollars into their schools, while many charter advocates have come to see a need for strong oversight in order to protect their brand from crooks and incompetents.
All three groups share the goal of wresting public tax dollars away from the public school system. But they differ in what should happen with those dollars once the wresting is done.