Which type of applications run primarily on remote Internet servers?

Web applications are primarily designed to run on remote Internet servers. They are accessed through a web browser and utilize the resources of a remote server to deliver services, store data, and execute processing tasks, rather than being installed on a user's local device. This allows for features such as multi-user accessibility from different devices, automatic updates, and centralized data management.

In contrast, shareware typically refers to software that is distributed on a trial basis, allowing users to evaluate it before purchasing, and often requires installation on local machines. Vertical-market applications are specialized software designed for specific industries or market niches and can run on local servers or devices. Public-domain software is software for which no ownership rights are claimed and is often available for free, but it may also run locally rather than on remote servers. Thus, web applications uniquely align with the description of running primarily on remote Internet servers, making this the correct answer.