MAMP stands for Mac Apache MySQL and PHP. Mac is the operating system, Apache is the web server, MySQL is the relational database management system, and PHP is the object orientated scripting language.
WAMP stands for Windows, Apache, MySQL and PHP. Each means the same as with MAMP, but WAMP allows websites to be built offline using a windows platform rather than a Mac platform.
When each of the four components come together, they create a local web server on your computer only, meaning you can build entire websites and see what they would look like online without being connected to the Internet.
MAMP/WAMP allows you to run a WordPress site on your computer only (without anyone else being able to see it because it is not actually on the internet), which is why they are good for testing out new features on an existing site, making bigger changes, or building an entire site from scratch.
Develop and design your site locally
Work in a faster development environment.
Use the local install as a partial backup of your existing site.
Develop sites in a secure environment. Due to the site being offline, it makes it very hard to hack.