What does it usually mean when dense spots are found in a mammogram?

Depends. Breast density on a mammogram can have a lot of meanings. It is a normal and natural finding in younger women, and does not have much meaning. Density can make the mammogram more difficult for the radiologist to read, and they may recommend other tests too, such as an ultrasound. A new density, when comparing a past mammogram to a new one, may mean further evaluation is needed.
Dense spots. If it is on a screening mammogram, it may potentially be abnormal, and extra views, and possibly ultrasound, are usually done to clarify what it is. Keep in mind that most breasts have many dense spots normally, so it is the radiologist's job to find the dense spot that doesn't belong.
Uncertain. Dense spots on mammogram is a vary vague term. It depends on the particular appearance of these dense spots. Dense spots could range anywhere from completely benign in appearance to highly suspicious for malignancy.