There are several ways to edit alerts when using Splunk Web. For example, within an app, you can click on Alerts and then from the Alerts page, you can click on Edit.
From the drop-down menu, you can select Edit Actions, as shown in the following screenshot:
Editing the description
Clicking on Edit Description shows you the alert name (title) and allows you to edit the description of the alert:
Clicking on Edit Permissions ...