a1  b1  a2  b2
 1   2   3   4
 5   6   7   8

c   d
1   2
5   6
3   4
7   8

import pandas as pd
import numpy as np

df = pd.DataFrame(np.arange(1, 9).reshape((2, 4)),
        columns=["a1", "b1", "a2", "b2"])

part1 = df.iloc[:,0:2]
part2 = df.iloc[:,2:4]

new_columns = ["c", "d"]
part1.columns = new_columns
part2.columns = new_columns

print pd.concat([part1, part2], ignore_index=True)

c  d
0  1  2
1  5  6
2  3  4
3  7  8

import pandas as pd
df = pd.DataFrame({'a1' : pd.Series([1,5]), 'b1' : pd.Series([2,6]), 'a2' : pd.Series([3,7]), 'b2' : pd.Series([4,8])})

df1 = df[['a1','b1']]
df2 = df[['a2','b2']]
df1.columns = ['c','d']
df2.columns = ['c','d']
df1.append(df2)

# Make data as in previous answers
import pandas as pd
import numpy as np

df = pd.DataFrame(np.arange(1, 9).reshape((2, 4)),
                  columns=["a1", "b1", "a2", "b2"])

# Melt both columns and concatenate
df = pd.concat([
    df[['a1', 'a2']].melt(value_name='c'),
    df[['b1', 'b2']].melt(value_name='d')],
    axis=1)

# Discard unwanted columns melt creates
df = df[['c', 'd']]