A vaccine passport is a currently hypothetical document that shows that a person has received a vaccine -- at this time, a Covid-19 vaccine -- that grants them rights to travel or participate in commercial activities within their community, city, state, or country.
This is akin to a number of existing systems which require vaccines for traveling between countries; while the best known international standard is the International Certificate of Vaccination or Prophylaxis, commonly called a Yellow Card, most doctors can hand you a piece of paper that will be accepted by the appropriate authorities to allow you to Officially Document that you can enter a given region. Because the framework for requiring and documenting vaccination exists, is well established, has been in use for decades, and requires willful ignorance on an absolutely inane scale to ignore, anyone talking about a 'vaccine passport' is not talking about anything like an actual passport.
What they are talking about is pretty much up in the air. As of this posting, some government officials are talking about various vaccination requirements for doing various things; any sort of documentation of the vaccination involved in one of these things will be dubbed a vaccine passport. This results in a meaningless term for people to have Strong Opinions about.
Currently, it looks like vaccine passports will usually be free mobile apps that people can download to certify that they have received a vaccine. If they feel that it benefits them to demonstrate this, they can. If you have not been vaccinated, or do not have a mobile device, this might limit you from doing certain things -- entering certain businesses, traveling on certain transports, attending a friend's birthday party... the possibilities are endless.
While this could be a bad thing, it is somewhat telling that the same people who argue for strong voter ID laws and border control are arguing against vaccine passports, and vice versa; and also everyone is carrying trackable and hackable computers hooked up to a public network in their back pocket all day, every day. This is not so much a privacy issue or a individual rights issue as it is a way for people to signal their ingroup, and as such we, as a species, are doomed. Not because of this, but because everything is like this.