I'm building .apk file for my Android application using Android tools - Export...
as guided here
My question is what are signed and unsigned .apk file?
How can we use usngined .apk file? (I tried to export an unsigned one but I cannot install it for my Galaxy Nexus)
Signing Android apps is just like signing any other application; it provides the user with some level of assurance that the code hasn't been tampered with since you released it.
Unsigned apps should be usable generally but do require lowering the security level. A quick Google search found a number of articles that explained that all you have to do is check the Unknown Sources
box in Settings->Security
.