You can use rename_axis, for removing set to None:

Copyd = {'Index Title': ['Apples', 'Oranges', 'Puppies', 'Ducks'],'Column 1': [1.0, 2.0, 3.0, 4.0]}
df = pd.DataFrame(d).set_index('Index Title')
print (df)
             Column 1
Index Title          
Apples            1.0
Oranges           2.0
Puppies           3.0
Ducks             4.0

print (df.index.name)
Index Title

print (df.columns.name)
None

The new functionality works well in method chains.

Copydf = df.rename_axis('foo')
print (df)
         Column 1
foo              
Apples        1.0
Oranges       2.0
Puppies       3.0
Ducks         4.0

You can also rename column names with parameter axis:

Copyd = {'Index Title': ['Apples', 'Oranges', 'Puppies', 'Ducks'],'Column 1': [1.0, 2.0, 3.0, 4.0]}
df = pd.DataFrame(d).set_index('Index Title').rename_axis('Col Name', axis=1)
print (df)
Col Name     Column 1
Index Title          
Apples            1.0
Oranges           2.0
Puppies           3.0
Ducks             4.0

print (df.index.name)
Index Title

print (df.columns.name)
Col Name

Copyprint df.rename_axis('foo').rename_axis("bar", axis="columns")
bar      Column 1
foo              
Apples        1.0
Oranges       2.0
Puppies       3.0
Ducks         4.0

print df.rename_axis('foo').rename_axis("bar", axis=1)
bar      Column 1
foo              
Apples        1.0
Oranges       2.0
Puppies       3.0
Ducks         4.0

From version pandas 0.24.0+ is possible use parameter index and columns:

Copydf = df.rename_axis(index='foo', columns="bar")
print (df)
bar      Column 1
foo              
Apples        1.0
Oranges       2.0
Puppies       3.0
Ducks         4.0

Removing index and columns names means set it to None:

Copydf = df.rename_axis(index=None, columns=None)
print (df)
         Column 1
Apples        1.0
Oranges       2.0
Puppies       3.0
Ducks         4.0

If MultiIndex in index only:

Copymux = pd.MultiIndex.from_arrays([['Apples', 'Oranges', 'Puppies', 'Ducks'],
                                  list('abcd')], 
                                  names=['index name 1','index name 1'])


df = pd.DataFrame(np.random.randint(10, size=(4,6)), 
                  index=mux, 
                  columns=list('ABCDEF')).rename_axis('col name', axis=1)
print (df)
col name                   A  B  C  D  E  F
index name 1 index name 1                  
Apples       a             5  4  0  5  2  2
Oranges      b             5  8  2  5  9  9
Puppies      c             7  6  0  7  8  3
Ducks        d             6  5  0  1  6  0

Copyprint (df.index.name)
None

print (df.columns.name)
col name

print (df.index.names)
['index name 1', 'index name 1']

print (df.columns.names)
['col name']

Copydf1 = df.rename_axis(('foo','bar'))
print (df1)
col name     A  B  C  D  E  F
foo     bar                  
Apples  a    5  4  0  5  2  2
Oranges b    5  8  2  5  9  9
Puppies c    7  6  0  7  8  3
Ducks   d    6  5  0  1  6  0

df2 = df.rename_axis('baz', axis=1)
print (df2)
baz                        A  B  C  D  E  F
index name 1 index name 1                  
Apples       a             5  4  0  5  2  2
Oranges      b             5  8  2  5  9  9
Puppies      c             7  6  0  7  8  3
Ducks        d             6  5  0  1  6  0

df2 = df.rename_axis(index=('foo','bar'), columns='baz')
print (df2)
baz          A  B  C  D  E  F
foo     bar                  
Apples  a    5  4  0  5  2  2
Oranges b    5  8  2  5  9  9
Puppies c    7  6  0  7  8  3
Ducks   d    6  5  0  1  6  0

Removing index and columns names means set it to None:

Copydf2 = df.rename_axis(index=(None,None), columns=None)
print (df2)

           A  B  C  D  E  F
Apples  a  6  9  9  5  4  6
Oranges b  2  6  7  4  3  5
Puppies c  6  3  6  3  5  1
Ducks   d  4  9  1  3  0  5

For MultiIndex in index and columns is necessary working with .names instead .name and set by list or tuples:

Copymux1 = pd.MultiIndex.from_arrays([['Apples', 'Oranges', 'Puppies', 'Ducks'],
                                  list('abcd')], 
                                  names=['index name 1','index name 1'])


mux2 = pd.MultiIndex.from_product([list('ABC'),
                                  list('XY')], 
                                  names=['col name 1','col name 2'])

df = pd.DataFrame(np.random.randint(10, size=(4,6)), index=mux1, columns=mux2)
print (df)
col name 1                 A     B     C   
col name 2                 X  Y  X  Y  X  Y
index name 1 index name 1                  
Apples       a             2  9  4  7  0  3
Oranges      b             9  0  6  0  9  4
Puppies      c             2  4  6  1  4  4
Ducks        d             6  6  7  1  2  8

Plural is necessary for check/set values:

Copyprint (df.index.name)
None

print (df.columns.name)
None

print (df.index.names)
['index name 1', 'index name 1']

print (df.columns.names)
['col name 1', 'col name 2']

Copydf1 = df.rename_axis(('foo','bar'))
print (df1)
col name 1   A     B     C   
col name 2   X  Y  X  Y  X  Y
foo     bar                  
Apples  a    2  9  4  7  0  3
Oranges b    9  0  6  0  9  4
Puppies c    2  4  6  1  4  4
Ducks   d    6  6  7  1  2  8

df2 = df.rename_axis(('baz','bak'), axis=1)
print (df2)
baz                        A     B     C   
bak                        X  Y  X  Y  X  Y
index name 1 index name 1                  
Apples       a             2  9  4  7  0  3
Oranges      b             9  0  6  0  9  4
Puppies      c             2  4  6  1  4  4
Ducks        d             6  6  7  1  2  8

df2 = df.rename_axis(index=('foo','bar'), columns=('baz','bak'))
print (df2)
baz          A     B     C   
bak          X  Y  X  Y  X  Y
foo     bar                  
Apples  a    2  9  4  7  0  3
Oranges b    9  0  6  0  9  4
Puppies c    2  4  6  1  4  4
Ducks   d    6  6  7  1  2  8

Removing index and columns names means set it to None:

Copydf2 = df.rename_axis(index=(None,None), columns=(None,None))
print (df2)

           A     B     C   
           X  Y  X  Y  X  Y
Apples  a  2  0  2  5  2  0
Oranges b  1  7  5  5  4  8
Puppies c  2  4  6  3  6  5
Ducks   d  9  6  3  9  7  0

And @Jeff solution: