Malware - short for malicious software
Any software designed with malicious intent purely to screw up your computer.
Adware - short for advertisment software
Software designed with intent to advertise prodtucts. Can often install secretly in the hope that you don't notice it happening. Can have a spyware component to provide you with "targeted" advertising through observing your browsing habits etc.
Spyware - spy software
Not used by 007. It's software that almost always installs secretly. It's purpose is to collect information on your browsing habits and or other computer uses and report this back to a third party (a person, a server, or another piece of software - usually adware. Can be a component of a peice of adware.