How can I "trash" or delete emails?

I will assume that you are using the Mail app. You will need to go through a process to determine where the problem may be. First, I would restart the computer in the Safe Mode. Hold down the Shift key on restart. Be patient as the startup process can take up to ten minutes while the operating system is repairing itself and emptying its caches. Then try moving some emails to the Trash. If that works, restart the computer normally and try again. If that still doesn't fix the problem you may then have to first move any saved emails to a newly created folder down under Local Folders. Then delete the email apps account and create a new account. If the problem still persists, then you may have to either (1) reinstall the email app if it is other than Mail, or (2) reinstall the OS to also reinstall the Mail app. I hope one of those things works for you.

First, please give us more specifics as to what you do and what the results are. When did this begin? Was it after changing anything? In order to evaluate and respond, we need to know what is happening.