How to make Intellij show Eclipse like API documentation on mouse hover

You can edit the idea.properties found in the bin folder of IntelliJ application and add this line:

auto.show.quick.doc=true

This will give you the JavaDoc when hovering over a variable:

Annoyed over the documentation poping up, but still like it?

In the begining it feels like a nice feature, but after a while I notice that the documentation box is in the way, therefore I disable it. Instead I use the shortcut keys when ever I want to know more about something.

To do so, place the caret on the class or method that you want to know more about, and press ctrl + Q and the documentation will be shown. Also, if you want to go to that class, you can press ctrl + B, and you can see the soure code, if you need it.

To show the documentation info with "Mac OS X 10.5+" keymap press CTRL + J (thanks @GreenTurtle for sharing this)

I recomend that learning to navigate with the keyboard, instead of using the mouse, it will save time and help you keep your focus.

This is not what the OP asked for, but I think it is a better alternative.

Simple docs will appear if you hold ctrl when you hover. To make proper docs appear just on hover (like eclipse) in IntelliJ v14.0:

File > Settings... > Editor > General > [Other]

Then at the very bottom there is an option: "Show quick documentation on mouse move":

The docs will now appear on hover:

Settings -> Editor -> "Show quick doc on mouse move"