There really isn't a hard and fast job description. Many mainly serve civil citations. Some handle animal control. Others have some professional SWAT trained warrant serving teams.
Since Constables are elected officials, county governments have limited control over their offices. In many areas the constable's office has a bad reputation. Others are quite good. It depends on the character of the individual Constable. Deputy constables are often part time and always badly paid.
In Dallas, after a bunch of official corruption scandals the Sheriff's office has taken over some of the constables' duties. In some other places that needs to happen.
In your case, what the local PD was really saying was "we are too busy with rapes, robberies and murders to deal with your complaint today and you will need to handle things yourself by suing the bad guy in JP court." Constables will serve the process and execute on any judgment you get. But, if you're lucky, and I mean really lucky, and there was something criminally fraudulent about what happened, the Constable may take an interest in your case, work it up, and present it to the DA. Don't hold your breath.