Hi, I gift itself would be called a "Birthday Gift". For example: I bought Sam a birthday gift I bought Nicola a Christmas gift If you are talking about the gift and explaining who you bought it for you would then say birthday gift for (and the person's name). For example: This is a birthday gift for Sam This is a Christmas gift for Nicola If you are talking about who possesses the gift (the owner) then you would use (person's name)'s birthday gift. For example: This is Sam's birthday gift This is Nicola's Christmas gift You can also use the expression "gift for" to specify the occasion that the gift has been bought for. For example: This is a gift for Sam's birthday This is a gift for Christmas (in this case Christmas isn't specific to a single person so you don't use the persons name) Gift is more common in American English. Present is commonly used in British English. I hope that helps, but please let me know if you have any other questions. Thanks, Sam

You could say: This is a birthday gift. This is a birthday gift for you. This is a gift for your birthday. I hope that helps.