Been throwing up for the last nine hours and now I have red spots around my eyes. What are they?

Petechiae. It sounds like they may be something called petechiae. They are very small bruises, caused by a trauma such as hard coughing or vomiting. In this case they are commonly seen around the eyes. They usually go away after several days. If they appear elsewhere on the body, get bigger, or are associated with a significant illness, then you should see your doctor.
Spots around eyes. They are caused by rupture of minute blood vessels following vomiting or coughing.