Ok, a few things could be going on. But let's get one thing straight--any user should be able to delete a knowledge object (such as a dashboard, saved search, or tag) that they have created but have not shared. Things get tricky when you share your knowledge objects with others or need to share or delete knowledge objects that others have created.
First, you can only delete knowledge objects that have been shared or that others have created if your role has write permissions for the app to which they belong. So for example, to delete a dashboard that you didn't create or share that belongs to the Search app you'd need write permissions for that app.
If you have the Admin role you should be able to delete objects in the search app, but if you have another role that has "equivalent capabilities" to the Admin role you won't be able to delete Search app objects that you didn't create or share unless your role has been granted write permission for the Search app.
To do this, you or your admin need to navigate to Manager > Apps and click Permissions for the app to go to its Permissions page. On this page you or your admin can determine which roles have read or write access to the knowledge objects that the app contains. When your role has write permissions you can delete these knowledge objects. Write permissions also dictate which roles can set permissions for knowledge objects such as dashboards--if you don't have this ability that's a good indicator that you don't have write permission at the app level for your role.
Second, you cannot delete default knowledge objects that were delivered with Splunk (or with the App) via Manager. If the knowledge object definition resides in the app's default directory, it can't be removed via Manager. It can only be disabled (by clicking Disable). Only objects that exist in an app's "local" directory are eligible for deletion.