It’s about giving parents a voucher (like a gift card) to spend on a school. The idea is that instead of subjecting everyone to a one size fits all public school system, which can have various problems and inefficiencies, you give parents money that can only be spent on schools but let them choose which provider to spend it on (whether public or private or homeschooling or whatever). It introduces an element of competition between schools but also choice, and the ability for parents to seek out what is best for their child. For example: teacher union avoiding accountability via test scores? Switch schools. School policies not properly managing disruptive students? Switch schools. Curriculum not appropriate for your child? Switch schools.
Usually a government program to give parents fixed annual tuition voucher which they can use to enroll their child at a private school. The voucher may cover part or all of the tuition at the private school and the parent would pay the remainder if there is one. It often goes under other monikers such as School Choice. Parents usually don't have much say as to which public school their child goes to, as it is assigned by address. These programs aim to let parents who are in the district of a poorly performing school to transfer their kids to a school they think will benefit them. Wealthier parents already have this option, but private schools are too expensive for even most middle class parents.
It is often related to efforts to create charter schools which are essentially private schools funded by taxes but run by a commercial or non-profit management company. These schools are usually regulated similarly to other private schools, having fewer regulations than the public schools and not under any direct authority of elected officials.
"Vouchers" are effectively tuition money for a school of a parent's choice, funded by the government. They would in theory allow parents to choose the "right" kind of schooling for their child, rather than having their child attend a school based on where they live.
This system does not exist today, but there are many who would like to see it. Lots of people are fed up with what they perceive as the failings of public schools. Very few who advocate for vouchers seem to advocate for education funding in general - most seem to simply want their kids somewhere else.
Historically, the push for vouchers is a right-wing, small-government kind of effort intended to further undermine the public school systems, and also to allow government funds to support religious education which, under the current system, is not permitted.
