Definition
A static IP is an IP address permanently assigned to a particular device, network, or account, and does not change over time. Most consumer ISP connections use dynamic IPs that may change at any reboot; static IPs are typically a paid upgrade. In the VPN context, a static IP is one that a specific subscriber receives every time they connect — it is, in effect, a dedicated IP that is also stable.
Static vs Dedicated
These terms are sometimes used interchangeably and sometimes distinguished. A dedicated IP is reserved for one user but might still rotate occasionally. A static IP, more strictly, is the same address every connection. In most VPN provider product pages, a 'dedicated IP' add-on is in fact also static — the same address comes back every time you connect. Some providers offer cheaper non-static dedicated IPs that are reserved per session but may change.
Use Cases
Static IPs are useful for whitelisting (your office allows access from one specific IP), for hosting a service reachable from the public internet, for reducing security challenges (banking and admin systems flag IP changes as suspicious), and for any application where a stable endpoint matters. They are the opposite of an anonymity feature — by definition, the IP is permanently linked to the subscriber.