An accu-chek is a company that makes blood testing meters.
ACCU-CHEK Blood Glucose Monitoring Systems | ACCU-CHEK
A fasting BG test is one performed after fasting, typicaly first thing in the morning. It may be done at a lab or with our personal meter, say an Accu-Chek Aviva. It gives an idea how your basals are being managed as opposed to food. Basal is the sugar continuosly being released by your liver.