If a person sends threatening/abusive/offensive messages to another person via Facebook, Twitter, or any other social networking site, they could be committing an offence. The most relevant offences are 'harassment' and 'malicious communications. The offence of 'stalking' may also be relevant if a person is being targeted persistently by someone across various social media accounts, along with any other repeated, obsessive and intrusive behaviour, that causes the person alarm and distress. See Q151 and the links in Related Information for further details.
For harassment to be committed, there must be a 'course of conduct' (i.e. two or more related occurrences). The messages do not necessarily have to be violent in nature, but must be oppressive and need to have caused some alarm or distress. See Q497 for further information regarding this offence.
If there has only been a single communication, which would be insufficient for the offence of harassment (above), there could be an offence relating to malicious communications. For such an offence to be committed, a message must be sent to another person (or sent via a public communications network) that is indecent, grossly offensive, obscene or threatening/menacing.
You can report any possible offences to your local police force, who will advise whether they can progress the matter based on the full facts and your individual situation. In order to assist the police with their investigation, you must not respond to the message as it may encourage the sender and make the situation worse. Also, you could take a screenshot of the message so if it gets deleted later there will still be a record of what was said.
However, depending on the circumstances and the nature of the messages, you may wish to initially make a report to Facebook/Twitter etc., as they have processes in place for such situations, and may be able to simply remove the content and/or close down the person's account.
Please see the links in Related Information for Facebook and Twitter's Community Guidelines in relation to dealing with harassment.