Vouchers are basically scholarships given to students so that they can go to a private school instead of their local public school. The arguments for vouchers are many. For one thing, people argue that schools will be forced to raise their standards in order to retain students if enrollment begins to decline. Further, vouchers provide lower-income students the opportunity to attend private schools that they might not be able to afford. Vouchers give these students more choice and control over their education.
The arguments for vouchers rely at least partly on the belief that public schools are failing. However, some people believe that the schools themselves are not the problem — it's the lack of support from community and parents for teachers and the schools that is really at issue. One reason that many private schools succeed where public schools fail is their greater incidence of parental involvement with students and school activities. Just giving students vouchers would not fix the problems themselves.
There is also a concern that if the state gives students vouchers to attend private schools, it might be seen as supporting religion. In other words, if a student chooses to use public funds to go to a Christian school instead of public school, taxpayers are paying for that student's education, which will likely include religious instruction. Whether or not this violates the First Amendment to the Constitution depends on who is reading the Constitution. Ultimately, the Supreme Court will be the judge of this issue.