Coolify is an open-source and self-hostable alternative to Heroku, Netlify, and Vercel. Docker Compose is a tool that simplifies the control of multi-container applications by defining and running them in a single YAML configuration file. Strapi is a headless CMS used to develop websites, mobile applications, eCommerce sites, and APIs without knowing anything about the backend or databases. To deploy a Strapi project on Coolify using Docker Compose, you need to install Coolify, add environment variables, select the repository, choose the Docker Compose file, and then click the "Deploy" button. The process involves copying essential folders and files needed to build the image, managing volumes, and ensuring proper functioning of the repository's code. However, challenges may arise such as missing dependencies, volume management, and network configurations, requiring manual intervention to resolve them.