ASP is an acronym which has become more confusing of late due to two disparate meanings.
The first, "Active Server Pages" is an authoring environment. ASP is commonly used for developing databases.
"Application Service Provider" is the term given to companies that rent or lease software (often across the internet). Such software ranges from web specific (log analysis tools, search engine promotion) to general business and accounting packages.
Atomz (site search tool) is an example of an Application Service Provider.