Pumpkin spice, sold as pumpkin pie spice, is a blend of four ground spices: cinnamon, ginger, nutmeg and allspice. McCormick, which has sold its pumpkin pie spice since 1934, lists exactly those four on the label, with cinnamon and ginger giving the blend most of its warm, spicy-sweet flavor.
Despite the name, the standard blend contains no pumpkin. It is called pumpkin spice because it was made to season pumpkin pie, and the same mix is now used in lattes, muffins, cookies and savory dishes such as sweet potatoes and squash.
Source: McCormick: Pumpkin Pie Spice product page and ingredient list
Image: McCormick Pumpkin Pie Spice, by Stephen Witherden, CC BY 2.0, via Wikimedia Commons.