If you want to use VPN to provide privacy, then only a ‘“no logs” provider will do. Unfortunately, when a provider claims to keep no logs, we just have to take its word for it (which is why the Edward Snowden’s of this world prefer to use Tor).
VPN providers have built their business model on promising privacy, and if it becomes known that they failed to do this (for example by keeping logs even when they promised not to, and then being compelled to hand these over to the authorities), their businesses would be worthless (and they might find themselves liable for legal action by the compromised individual).
In addition to keeping no logs, any company that cares about protecting their users’ privacy also uses shared IPs. This means that many users are assigned the same IP address, so matching identified internet behavior with a specific individual is very difficult to do, even if a provider should wish (or is compelled) to do so.
More privacy-minded VPN companies allow you pay for their services anonymously. The most common method is using Bitcoins**, but companies such as Private Internet Access will accept anonymously purchased store cards, and Mullvad will even take cash sent by post!
This adds an extra layer of privacy, as the VPN company does not know your real name, address, or banking details. It will, however, still know your real IP address*