Santa Claus is as much a part of Christmas as Jesus Christ. Some people, in fact, celebrate Santa Claus more than the birth of Jesus Christ.
One thing that should be known is that Christmas is primarily about the birth of Christ, and not about the ritual of distribution and exchange of gifts. Also, there is no relationship between Christ and Santa Claus as such.
The ritual of distributing gifts on Christmas started long after the birth of Christ, and was started in a good faith by the real Santa Claus, named Saint Nicholas. St. Nicholas was a devout believer of Jesus and became a priest at the age of 17.
(Courtesy: Catholic Online)
St. Nicholas was born after 280 years of Christ’s death, in Turkistan. Being an orphan he was brought up in a monastery. He was kind-hearted and humble right from the start. He was always very keen to help the needy and used to distribute gifts to Children quite a lot.
His kind-heartedness made people believe that St. Nicholas will keep giving gifts even after his death. So it is this belief that started the ritual of gift distribution on Christmas.
He used to visit people at night and leave clothes, food, etc. for them. After his death, people started dressing up like him and distributing gifts.