Name:   ID:       emails:
apple   372       frt@apple.com
yandex  295      drft@yandex.com
mail    215      sdrf@mail.ru
king    352       king@king@com
mac     097       srt@mac.com
yandex  295     info@yandex.com
mail    215     ivanov@mail.ru

Name:   ID:       emails:
    apple   372       frt@apple.com
    yandex  295      drft@yandex.com,info@yandex.com
    mail    215      sdrf@mail.ru, ivanov@mail.ru
    king    352       king@king@com
    mac     097       srt@mac.com

In [161]: (df.groupby(['Name','ID'])
             ['emails']
             .apply(','.join)
             .reset_index(name='emails'))
Out[161]:
     Name   ID                           emails
0   apple  372                    frt@apple.com
1    king  352                    king@king@com
2     mac   97                      srt@mac.com
3    mail  215      sdrf@mail.ru,ivanov@mail.ru
4  yandex  295  drft@yandex.com,info@yandex.com