For PyQGIS 3 this might be the easiest way to delete a single field:

# 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()

Or if you want to delete multiple fields at once, you can go for this slightly modified:

# Get a layer object:
my_vectorlayer = iface.activeLayer()
# Create a list of the fieldnames you want to delete:
fieldnames_to_delete = ['field_a','field_c']
# Enter Edit mode
with edit(my_vectorlayer):
    # Create empty list we will fill with the fieldindexes
    fields_to_delete = []
    # Get the field index by its name:
    for fieldname_to_delete in fieldnames_to_delete:
        fields_to_delete.append(my_vectorlayer.fields().indexFromName(fieldname_to_delete))
    # Delete the fields by their indexes, note that it has to be a list:
    my_vectorlayer.dataProvider().deleteAttributes(fields_to_delete)
# Update the fields, so the changes are recognized:
my_vectorlayer.updateFields()

For PyQGIS 3 this might be the easiest way to delete a single field:

# 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()

Or if you want to delete multiple fields at once, you can go for this slightly modified:

# Get a layer object:
my_vectorlayer = iface.activeLayer()
# Create a list of the fieldnames you want to delete:
fieldnames_to_delete = ['field_a','field_c']
# Enter Edit mode
with edit(my_vectorlayer):
    # Create empty list we will fill with the fieldindexes
    fields_to_delete = []
    # Iterate over the list of fieldnames and get the indexes
    for fieldname_to_delete in fieldnames_to_delete:
        # Get the field index by its name:
        fieldindex_to_delete = my_vectorlayer.fields().indexFromName(fieldname_to_delete)
        # You can also check if the field exists
        if fieldindex_to_delete == -1:
            # If it does not exist, just skip it and go to the next one. This may prevent a crash or error :)
            continue
        # Append the index to the list
        fields_to_delete.append(fieldindex_to_delete)
    # Delete the fields by their indexes, note that it has to be a list:
    my_vectorlayer.dataProvider().deleteAttributes(fields_to_delete)
# Update the fields, so the changes are recognized:
my_vectorlayer.updateFields()

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()

For PyQGIS 3 this might be the easiest way to delete a single field:

# 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()

Or if you want to delete multiple fields at once, you can go for this slightly modified:

# Get a layer object:
my_vectorlayer = iface.activeLayer()
# Create a list of the fieldnames you want to delete:
fieldnames_to_delete = ['field_a','field_c']
# Enter Edit mode
with edit(my_vectorlayer):
    # Create empty list we will fill with the fieldindexes
    fields_to_delete = []
    # Get the field index by its name:
    for fieldname_to_delete in fieldnames_to_delete:
        fields_to_delete.append(my_vectorlayer.fields().indexFromName(fieldname_to_delete))
    # Delete the fields by their indexes, note that it has to be a list:
    my_vectorlayer.dataProvider().deleteAttributes(fields_to_delete)
# Update the fields, so the changes are recognized:
my_vectorlayer.updateFields()