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

In [24]: df
Out[24]:
     Name   ID           emails  phone
0   apple  372    frt@apple.com  +1111
1  yandex  295  drft@yandex.com  +2222
2    mail  215     sdrf@mail.ru  +3333
3    king  352    king@king@com  +4444
4     mac   97      srt@mac.com  +5555
5  yandex  295  info@yandex.com  +6666
6    mail  215   ivanov@mail.ru  +7777

In [25]: (df.groupby(['Name','ID'])
    ...:    [['emails','phone']]
    ...:    .agg(','.join)
    ...:    .reset_index())
    ...:
    ...:
Out[25]:
     Name   ID                           emails        phone
0   apple  372                    frt@apple.com        +1111
1    king  352                    king@king@com        +4444
2     mac   97                      srt@mac.com        +5555
3    mail  215      sdrf@mail.ru,ivanov@mail.ru  +3333,+7777
4  yandex  295  drft@yandex.com,info@yandex.com  +2222,+6666