So, in order to curb everyone joking around, here's a serious answer:
To do something recursively in this context means to apply whatever it is that you're doing to all the children of the object you're applying it to.
chown $HOME takes ownership of your home folder, but only the home folder, nothing inside it.
chown -R $HOME (note the upper-case R) will take ownership of the home folder, and then go into the home folder, find all the objects (files and folders in this case), take ownership of them, and then check to see if those objects have any children that it can take ownership of too. Repeat and rinse until it runs out of things to take ownership of.