The ability to delete discussions and/or comments is a permission typically reserved for Admins and Mods.
It is controlled with Roles and Permissions, and can be set on a per category level:
The following features are available from the dropdown menu on a discussion at the category and discussion level. This drop down gives option to Delete the discussion and to access deleted comments from target discussion.
I deleted a post accidentally! What can I do!?
You can restore it from the change log. Please note that items deleted forever from the change log cannot be recovered.
Can users delete their own discussions and comments?
In Vanilla's default set up, only an Administrator or a Moderator has the permission to delete a post (discussion and/or comment). A discussion may have many comments, and so deleting a discussion with many comments 'orphans' those comments. This will essentially deleting them.
You can have an option enabled in your config that would allow users to delete their own comments, this can be requested from your CSM or Vanilla Support.
How can users request to have a discussion deleted?
Some communities will create a delete option using the Reported Posts add-on, for instance 'Please delete this discussion', to allow users to request that their post be deleted by the community management team. This will allow your team to assess whether the discussion has many comments, and if so perhaps it is preferable to edit (or blank) the discussion and change the author rather than delete the post outright.