MEAN stands for MongoDB, Express.js, AngularJS, and Node.js. This is a relatively new "stack" of software for delivering scalable cloud-hosted applications. The MEAN Stack is considered a more "modern" version of the LAMP Stack.
A stack is a collection of software technology that is packed together to form a platform. A stack is like a collection of tools that can be used to solve a problem. In this case, a MEAN stack solves a recurring problem in programming: how do I securely host web applications in the cloud?
MongoDB is an open source, nonrelational database management system (DBMS) that uses flexible documents instead of tables and rows to process and store various forms of data]. Mongo is a distributed database at its core, so it scales well in a cloud environment, and stores data in JSON-like formatted documents.
ExpressJS forms the MEAN stack's backend, handling all the interactions between the frontend (Angular) and the database (MongoDB).
Node.js is the backbone of the MEAN stack. Express and Angular are purpose-built to work on top of Node.js, making it easy to deploy your MongoDB database and application to the cloud.
MEAN is a newer solution to newer problems. MEAN exists to provide web services via the cloud, and take advantage of the cloud's hugely scalable technologies. It is also open source.