Gifts totaling more than $13,000 to one person in one year are considered a taxable gift and generate a potential Gift Tax. It does not matter if you give one $13,000 gift or 13 gifts of $1,000 each, or one gift of $11,000 and a "birthday gift" of $2,000. Gifts of a "future interest", no matter what their value, also are considered a taxable gift.
Gifts beyond the exemption limit (there is an exception for gifts that are directly paid by the gift giver for tuition and medical expenses) are considered "taxable gifts."
Taxable gifts generate a Gift Tax. But Gift Tax is not due until you give away over $1,000,000 in your lifetime.