VPNs can make your browsing private, but that doesn’t necessarily mean you’re anonymous. VPN services can and do log traffic (even the ones that say they don’t log do need to log some information, or they wouldn’t be able to function properly), and those logs can be requested by the authorities. Think of a VPN as being like curtains: people can’t peek through your curtains if you’ve got them closed, but curtains won’t hide your house.
What that means in practice is that VPNs are fine for bypassing geo-blocks, for protecting your online banking and for keeping business communications free from interception. However, if you’re using the internet to fight repressive regimes or to do anything else that could attract the attention of the authorities where you live, a VPN is not a magic wand that’ll make you invisible.