VPN's can be used for a number of different things, but for a normal consumer VPN, all it primarily does is hide your internet traffic from your ISP. For example, they will see the traffic as going to a VPN endpoint and won't know that it is going from there to wrongplanet.net.
One interesting thing is how good some sites are at figuring out who you are. Google, of course, pretty much knows who you are whether or not you are using a VPN. They can do this through things like browser fingerprinting and cookies. Also, if you have signed into gmail or some other service, they got you.
What surprises me is how much information Facebook knows about me. I use an alias that shouldn't tie to me at all since I don't care about the social media aspects -- I primarily use it to connect to local restaurants to find out their daily special. Facebook figured out pretty quickly that the alias I use on it is not my name, but they have narrowed it down so much that they have included a number of my relatives in the numerous suggestions for friends requests. A VPN doesn't seem to slow them down much at all.
A consumer VPN can increase privacy, but in a limited way. It doesn't make you invisible.
By the way, I'm doing more and more to distance my on-line presence from my real identity. When signing up at a web site, the web site gets a unique e-mail address that won't tie back to me. If you look at my signature, you will see an e-mail address. There is no way that I know of, other than hacking the involved networks, to tie that e-mail address to my real identity. I'm also using aliases more and more. On one recent web page, I posted two messages a few minutes apart with wildly different aliases, each with its own e-mail address.
If you really want privacy, you probably need to make it really difficult to connect your internet presence with your own identity. A VPN may help in a limited way.
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[email protected](The above is a valid, but temporary, e-mail address that is relayed to my real e-mail. If it starts getting spammed, I'll just remove it and create a new one and change the signature block. If you want to send me an e-mail, feel free to do so, but be aware that unless you are using something like this, your e-mail address will be exposed to me.)