Cloudfoundry open source software is the community maintained software and tools required to run cloud foundry on your own infrastructure. website: https://cloudfoundry.org.
Pivotal CF is a Pivotal branded installation of cloud foundry on your own infrastructure (AWS, VMware, OpenStack, Azure, etc) which has enterprise-grade support. It also offers an array of services like MySQL, Redis, ELK, RabbitMQ, etc. that can leverage their cloud foundry installation and have applications that use these services.