What does it mean if my cervix starts to open before I am full term?

Depends. Women who have had babies before often dilate before they go into labor. In first time moms early dilatation puts one at risk for early delivery, but one still may deliver at term. If the dilatation is happening after 36 weeks it is of litle concern. Dilatatation in the 24-34 weeks range is more concerning for being a precursor to preterm labor.
Dilation. Early dilation can be a sign of preterm delivery. If you have had previous children it can be normal (common). If you are very early in gestation, you need to see your doctor.