While your garden variety spam is a mere nuisance, harassing email can be upsetting and even scary. What makes it even scarier is the fact that the inherent anonymity of the internet makes it pretty difficult to find the identity of the person who is sending you threatening emails. This is not meant to suggest that you should ignore harassing email- on the contrary- you should save the email and promptly report it to the proper authorities. This includes your email provider, the sender's email provider, your local authorities, and anyone else in a position of authority who will listen to you.
A person who is willing to harass you via email is probably comfortable doing so because he or she feels safe from discovery. This person may know the way around the net well enough to use a false IP address and a completely bogus identity with the email provider. Some email providers are beefing up their personal information requirements so that anyone signing up for a new account must list a full name and a valid mailing address before they can get an email account. That said, a person could sign up for an email account using the full name 'Mickey Mouse' and a mailing address picked at random out of the phone book to get past these requirements. The only way to keep people from getting bogus email accounts would be to require verification of all listed information when signing up for an email account- and that would cost the email providers money, so free email accounts would probably become a thing of the past. This scenario could become a reality eventually, thanks to all the people who choose to abuse the email system.
While it is difficult if not impossible to find out on your own who is sending you harassing email, there are services you can turn to for help. Computer specialists and investigators can use tools and knowledge to track down an email abuser that private citizens don't have access to. For example, if you know how to look at the headers on an email, you can get the IP address of the internet provider, which can possibly narrow down the area of the country where the email is coming from- if the person is not using a fake IP. For the person who is receiving the harassing email, that doesn't help much. But people and businesses with the proper authority can trace the IP, contact the owner and get a hold of the records that could identify the person responsible for the harassment. Getting this information can require a court order- something the average citizen would have a hard time getting.
If you're getting harassing email, do not reply to it, and do not delete it. Save the email and contact the authorities listed above. You may get the fastest results by hiring an online detective company that can do a reverse email search and other services- and they can communicate with the proper authorities for you to make sure that the perpetrator is brought to justice.