CRM Online is a cloud-based offering of Microsoft Dynamics CRM where all the backend processes (such as application servers, setups, deployments, databases, licensing, etc.) are managed on Microsoft servers. CRM Online is a subscription-based offering which is preferred for organizations who may not want to manage all the technicalities involved in a CRM implementation. You can get started with setting up your system in a few days (not weeks, months or years) and access it on web via your browser.
CRM on-premise is a more customized and robust offering of Microsoft Dynamics CRM, where the CRM application and databases will be deployed on your servers. This offering allows you to control all your databases, customizations, deployments, backups, licensing and other network and hardware setups. Generally, organizations who want to go for a customized CRM solution prefer on-premise deployment as it offers better integration and customization capabilities..