Outlook rules: client-side vs server-side rules

You can manage rules by logging into Outlook Web Access (OWA).

In the OWA options section is where you will find the rules management page. This page allows you to create, edit, delete, and prioritize rules. Rules that have errors will be highlighted in red. Client side rules that require Outlook to complete an action will show up in gray.

Server-based rules If you have a Microsoft Exchange account, the server can apply rules to your messages even if Outlook is not running. These are called server-based rules. The rules must be set to be applied to messages when they are delivered to your Inbox on the server, and the rules must be able to run to completion on the server. For example, a rule cannot run to completion on the server if the action specifies that a message be printed. If a rule cannot be applied on the server, it is applied when you start Outlook.

Client-only rules A "client" is the e-mail program that runs on an individual's computer. A rule that cannot be applied by the server is a "client-only rule" because it runs only on the individual's computer. If your list of rules contains rules that can be run on the server as well as those that cannot, the server-based rules are applied first, followed by the client-only rules.

Reference link: Microsoft Exchange Team Blog