OK. This is how I got it to work. I first followed MoMo's advice, that is...
If the LogCat panel is empty in Eclipse the emulator doesn't have the
focus. Go to the DDMS perspective and try clicking on the 'emulator'
entry in the Devices panel (top-left screen).
But to no avail.
I then attempted to
reset adb (Android Debug Bridge) as suggested by furikuretsu.
How? Window -> Devices -> upside down triangle menu button -> Reset adb.
It also didn't work, but I did get the following message:
Android hierarchyviewer: Unable to get the focused window from device
This meant that MoMo was right in that my Android device or emulator didn't have focus. However, the solution I did in my case is different.
What worked for me:
1) Replugged my Android device, which was connected to my computer via USB.
2) Restarted Eclipse, as mentioned by Abu Hamzah (although since I didn't know I can do
File -> Restart, I manually closed down Eclipse, and then restarted the application again.)
I can now see logs in my logcat.
Hope this helps you too.