BGP (Border Gateway Protocol) is the protocol that runs routing between the large networks of the Internet. Each network (identified by an AS number) announces via BGP the IP prefixes it serves, and chooses which paths to reach others.
In practice
For a company, “running BGP” enables multi-homing: announcing its IP addresses via several carriers for resilience and automatic failover. It requires its own AS number and an address block (PA or PI).