Definition
A firewall is a hardware device or software process that enforces a set of policies on network traffic crossing its boundary. The simplest firewalls match against IP addresses and ports; modern Next-Generation Firewalls inspect protocols, application signatures, and even decrypt TLS to inspect content. Firewalls exist at every level of the network: in your operating system, on your home router, at corporate boundaries, and at national gateways.
Firewalls and VPNs
A VPN intersects with firewalls in two ways. From inside a restrictive network (workplace, school, country), a VPN can tunnel out past a firewall by encrypting traffic so the firewall cannot identify and block it. From the firewall operator's perspective, this is exactly what they were trying to prevent, which is why DPI and obfuscation exist as the next round of the arms race. On the protective side, a personal firewall on your device can be configured to drop all traffic that does not go through the VPN tunnel — effectively acting as a kill switch.
VPN Servers Behind Firewalls
VPN providers themselves run firewalls in front of their servers to drop malicious or out-of-spec traffic. Some advanced VPN clients let the user configure outbound firewall rules per-application, granting some apps internet access only through the tunnel and others only direct access — effectively a granular kill switch combined with split tunneling.