Spyware is software that installs on a computer system and is designed to collect information about the computer user, often without their knowledge or consent. This information may result in so-called identity theft, or theft of valuable information (such as bank or credit card details) or business data.
Rootkits are programs that install on your system, while keeping themselves, their processes, services and registry keys hidden, to stay invisible to the user. They represent a substantial security risk on home and company networks and are notoriously difficult to find and remove. Rootkits are normally deployed via another malware infection (eg a Trojan).