Firstly, try /r/Archaeology.

Secondly, it varies by country. Here in the UK, you need a permit from the Home Office before digging any area that is likely to contain human remains. If it's a surprise find, you call the coroner, and he decides if it's recent enough to call the police; if not, you apply for the permit, and carry on.

Source: Personal experience of excavating an Anglo-Saxon cemetery.

Further reading: https://www.google.co.uk/#q=human+remains+uk+law

As said before, r/archaeology would be glad to help, and probably will have more people with this experience. This depends on the country, the type of land its found on, and its age.

In the US, NAGPRA deals with human remains excavated on federal land. It was enacted to ensure that Native American tribes have the final say in what happens to remains and associated artifacts that might belong to that tribe or its ancestors. Check out the site for answers to specific questions about it.

From the archaeological perspective it largely depends on context. The excavation and data collection will be different whether its a single interment in an unexpected location or in a cemetery, whether its a open sepulcher, multiple burial or an in-ground one, or whether its a complete individual with tissues remaining or a single ulna in architectural fill.