Heroku (pronounced her-OH-koo) is a cloud application platform which lets you host your web application and deploy updates and changes to it over the internet. It can be used for small projects (like the ones we make here at One Month), or scaled for use by enterprise software companies.
That being said, Heroku is a fairly finicky tool, and chances are you will find yourself doing something that makes Heroku upset. Luckily, everyone else who uses Heroku has also made Heroku upset, and Googling for Heroku error messages will bring up many possible solutions. Oh, Heroku.
Heroku Dev Center - Heroku
This is a great place to learn the basics of deploying to Heroku, based on the language your application is written in.
Introduction to Heroku - Jason Hansen
A quick overview of getting started with Heroku.