Depending on the circumstances and the nature of the messages received, it is recommended that you initially make a report to the social media platform you are using (e.g. Facebook). They may be able to simply remove the content and/or close down the person's account.
If threats or abuse have been received or indecent images have been uploaded and you believe that an offence may have been committed, you can report this to your local police force. They will be able to confirm whether this is the case, based on the full facts and your individual situation.
The most relevant offences are 'harassment' and 'malicious communications'. For harassment to be committed, there must have been a clear 'course of conduct'. That is, two or more related occurrences. The messages do not necessarily have to be violent in nature, but would need to have caused some alarm or distress.
If there has only been a single communication, it's unlikely it would qualify as harassment, but could be considered a malicious communication. 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, threatening or menacing.
You can use the links below to visit various sites which may be useful to you, including how to report abuse on the various social media platforms, as well as advice from the police and Crown Prosecution Service.