Looks like a poor sheet vinyl installation. Why they ran it up the wall is anybody's guess. Probably so they wouldn't have to install base trim.
I would use a pry bar or a stiff putty knife to get in behind where it's attached to the wall and pry that off as best you can. Then you can see how the rest of the floor is attached. It may be glued down, or, if you're really lucky, it could be laid loose on the floor without adhesive. Odds are, it's glued down. About all you can do is pull it up to disengage the glue. Use your putty knife or if you have a floor scraper to get underneath it and scrape it off. It is usually quite stubborn and you may need a heat gun to soften the adhesive. The heat gun gets quite hot, so be very, very careful with it.
Sometimes it's glued down so well that you can't budge it off. In those cases, it's just better to put down some underlayment over top of it and install new vinyl.
I would use a pry bar or a stiff putty knife to get in behind where it's attached to the wall and pry that off as best you can. Then you can see how the rest of the floor is attached. It may be glued down, or, if you're really lucky, it could be laid loose on the floor without adhesive. Odds are, it's glued down. About all you can do is pull it up to disengage the glue. Use your putty knife or if you have a floor scraper to get underneath it and scrape it off. It is usually quite stubborn and you may need a heat gun to soften the adhesive. The heat gun gets quite hot, so be very, very careful with it.
Sometimes it's glued down so well that you can't budge it off. In those cases, it's just better to put down some underlayment over top of it and install new vinyl.