You can use melt for reshaping, then split column variable and drop and sort_values. I think you can cast column year to int by astype and last change order of columns by subset:
df1 = (pd.melt(df, id_vars=['county','area'], value_name="pop"))
df1[['tmp','year']] = df1.variable.str.split('_', expand=True)
df1 = df1.drop(['variable', 'tmp'],axis=1).sort_values(['county','year'])
df1['year'] = df1.year.astype(int)
df1 = df1[['county','year','pop','area']]
print (df1)
county year pop area
0 1001 2006 1037 275
3 1001 2007 1052 275
6 1001 2008 1102 275
1 1003 2006 2399 394
4 1003 2007 2424 394
7 1003 2008 2438 394
2 1005 2006 1638 312
5 1005 2007 1647 312
8 1005 2008 1660 312
print (df1.dtypes)
county int64
year int32
pop int64
area int64
dtype: object
Another solution with set_index, stack and reset_index:
df2 = df.set_index(['county','area']).stack().reset_index(name="pop")
df2[['tmp','year']] = df2.level_2.str.split('_', expand=True)
df2 = df2.drop(['level_2', 'tmp'],axis=1)
df2['year'] = df2.year.astype(int)
df2 = df2[['county','year','pop','area']]
print (df2)
county year pop area
0 1001 2006 1037 275
1 1001 2007 1052 275
2 1001 2008 1102 275
3 1003 2006 2399 394
4 1003 2007 2424 394
5 1003 2008 2438 394
6 1005 2006 1638 312
7 1005 2007 1647 312
8 1005 2008 1660 312
print (df2.dtypes)
county int64
year int32
pop int64
area int64
dtype: object