Names in India

Ganesha is known not by one name or two, but by many names, Each name has an explanation.

Because of his elephant face he is called Gajanana ('Gaja' meaning elephant);
As the leader of Shiva's Gana he is known as Gananayaka;
Ganesh As he is Leader of Shiva's Attendants,
Pillaiyar As The Noble Child, BenakaOne with a Long Nose ,
Gajamukha One with an Elephant's Face,
Vakratunda One with a twisted trunk,
Lambodara One with a large belly,
Surpakarna One with the pointed ears,
Vamanarupa One who is short in Stature, He bestows boons on his devotees and so is Varasiddhi Vinayaka,
He is Vidya Ganapati because he grants knowledge and wisdom;
He is called Gowriputra because he was born by the grace of Parvati,
Ekadanta because he has only one tusk, and
Vighneshwara because he is the Lord of all obstacles - so he has many names.

Names in Abroad

Vinayaksha or Kobo Daishi or Sho-Ten In Japan,
Kwanshidiyik or Kuan-Shit'ien or Ho-TeiIn china ,
Brahganesh or Prah-Kenes Cambodia Totkar or khaghan Mangolia,
Tchoprak or Ts'ogs-Bdag Tibet,
Maha-Pienne Burma,
Heramba Nepal,
Mahabini Borneo.

The belief that, Ganapati loves his devotees immensely is the reason for his worship everywhere.