Arrange the lollipop sticks in the lollipop molds and place on a baking sheet. Set aside.
Combine the sugar, water, and cream of tartar in a heavy-bottomed saucepan over medium heat. Stir until the sugar is dissolved. Increase the heat to high and, without stirring, cook until the sugar is amber or until a candy thermometer reads between 300 to 310 degrees F (hard-ball stage), about 10 minutes (see Cook's Note*). Promptly remove the saucepan from the heat. Carefully stir in the flavored extract.
Work quickly, drop 1 teaspoon of the hot mixture into each lollipop mold. Reheat the mixture briefly over low heat if it starts to thicken. Let the lollipops cool for at least 15 minutes. Remove from molds and store in airtight containers or wrap individually.
To determine hard-ball stage without a candy thermometer, remove the pot from the heat and dip the bottom of the pot in water to stop the mixture from cooking further. Carefully spoon a drop of the hot syrup mixture into ice-cold water. Remove the drop from the water after 5 seconds; when bent, it should snap easily and shouldn't be sticky to the touch. The cooled drop should have a yellowish tinge.