A VPN is simply an encrypted connection between two computers, each side running VPN software. The two sides, however, are not equal.
The software that you, as the user of a VPN service deal with, is known as the VPN client. The software run by a VPN company is a VPN server. The encrypted connection always starts with a VPN client making a request to a VPN server.
There are many different flavors of VPN connections, each with its own corresponding client and server software. The most popular flavors are probably L2TP/IPsec, OpenVPN, IKEv2 and PPTP.
Some VPN providers support only one flavor, others are much more flexible. Astrill, for example, supports OpenWeb, OpenVPN, PPTP, L2TP, Cisco IPSec, IKEv2, SSTP, StealthVPN and RouterPro VPN. At the other extreme, OVPN, as their name implies, only supports OpenVPN.