For PyQGIS 3 this might be the easiest way:
# Get a layer object and enter edit mode:
my_vectorlayer = iface.activeLayer()
with edit(my_vectorlayer):
# Get the field index by its name:
my_field = my_vectorlayer.fields().indexFromName('name of field you want to delete')
# Delete the field by its index, note that it has to be in a list:
my_vectorlayer.dataProvider().deleteAttributes([my_field])
# Update the fields, so the changes are recognized:
my_vectorlayer.updateFields()