In Brainstorm, only admins can remove ideas, pipelines, and challenges.   And they don't actually delete them, but simply remove them from view through "archiving."  Here's how it works.

On an idea page, pipeline page, or challenge page, look for the "show admin tools" link near the bottom of the right panel:

Next to "Archived" click "toggle":

Archiving an idea, challenge, or pipeline causes it to stop showing up in searches, lists, etc.  The content stays hidden in the database and can be later “unarchived” if the need arises.  If someone visits a page that has been archived (perhaps they follow a link from an old email), they see only a page with a message that the page has been archived, like this:

Why can't just anybody delete stuff in Brainstorm?  Well, first off, ideas, pipelines, and challenges quickly become community spaces that would make it unfair to allow a single individual to delete.  Also, some companies require that all of this type of data be retained, making deletion a legal no-no.