Correct spelling, explanation: the noun sponsor has its roots in the Latin word that was derived from the verb spondere, meaning to promise. First, it was used in 1651 in a religious sense. Considering the origin of the word, we certainly say that the only correct form is sponsor, not sponser.

Definition of sponsor:
noun, a person or a group that pays all or part of the expenses connected with a sport or artistic events, in return for advertising,
The company from our town is a sponsor of next week’s race.
John will be my sponsor and I can join the club.

Phrases with sponsor:
title sponsor – a company that buys naming rights to a building or a venue in order to increase its visibility and improve its reputation. In exchange for this, it provides money, goods, or services,
In our town, a natural gas company became the title sponsor for the football stadium.
corporate sponsor – a company that supports a person or organization by giving them money in order to promote its services and expand its reach,
He was disappointed when the company refused to be his corporate sponsor.

Incorrect spelling, explanation: there are two suffixes: or and er that are used to make nouns usually from verbs. But the word sponsor is not only a noun. It functions also as a verb. Besides, it stems from the Latin word sponsor, not sponser. Thus, sponser is spelled incorrectly and users of the language should memorize how it is written in a proper way. This is sponsor, not sponser.