Your email system on most servers are limited in available space. This is done to ensure that the entire server is not bogged down with to many stored messages. More importantly file attachments. Your host may limit the space you have available to store messages. You will no longer receive messages if you exceed your allocated drive space.
Your mail will exist on the server until purged. This unfortunately this is a multi-step situation. You delete the email then empty the trash.
Depending on your servers spam software, spam or junk. The mail reader you use, Trash or Archived.
You should on a Routine basis go though each of these directories and clear out the ones you no longer actually need. Some software such as Squirrel mail (one of the 3 default web readers on our system) have a system to by pass the trash bin. It also has a purge trash link to simplify the process.
Most readers you must go to the directory and Select all and delete. This moves them to the trash. Emptying the trash will delete them from the system permanently them. Deleting the trash also deletes the attachments as well. This is especially important if you send a lot of photos or files with your emails. A picture can eat a few megs a videos several gigs. Some mail systems will limit the maximum size of the attachment.