Web services, running through Internet Information Server, is what processes the ASP.Net and sends back the html to a user's browser. Application services is where the "brains" of SharePoint are located. All of the SharePoint specific logic and services are processed there. Database services are run by SQL Server and it's where all the data/content for SharePoint is stored.
All the services to run a SharePoint farm can be run on one server or on multiple servers. For scalability (performance) and redundancy (fail-safe) purposes a distributed (multi-server) architecture is highly recommended.
Single Server Farm