Make new column in Panda dataframe by adding values from other columns

Very simple:

df['C'] = df['A'] + df['B']

Building a little more on Anton's answer, you can add all the columns like this:

df['sum'] = df[list(df.columns)].sum(axis=1)

The simplest way would be to use DeepSpace answer. However, if you really want to use an anonymous function you can use apply:

df['C'] = df.apply(lambda row: row['A'] + row['B'], axis=1)

You could use sum function to achieve that as @EdChum mentioned in the comment:

df['C'] =  df[['A', 'B']].sum(axis=1)

In [245]: df
Out[245]: 
   A  B   C
0  1  4   5
1  2  6   8
2  3  9  12