When I build an apk on my own system, I can use adb install <filename> to install it and run it in the emulator. But when I download an apk, I get the following error:
EOCD not found, not Zip
file '<filename>' is not a valid zip file

Is this normal, or do I probably have something set up wrong? I would like to run some apps through the emulator because I don't own a device.

Check the apk of the file you downloaded, can you open it in a program like winzip without it saying "it's corrupt" as I believe that may be your problem

You're right. I downloaded a different one and it installed just fine. What could be the problem with the apk that didn't work? Did the developer do something wrong? I have heard of lots of people having this problem, so it is somewhat common.

It'll just be a corrupt download either between you and the server or the server and the uploader. Happens alot.