How can you tell when someone goes into cardiac arrest?

No pulse. Cardiac arrest as the name implies is when the heart stops. When your heart stops beating, blood carrying oxygen does not flow to your body and brain. The body ceases all functions (you stop breathing, you are unconscious, the skin turns pale or blue). The best way to tell if someone is in cardiac arrest is to use a heart monitor (like carried by ems personel) to monitor activity of the heart.
No pulse. That is the hallmark of cardiac arrest. The heart goes into ventricular fibrillation which is a heart rhythm that does not permit the blood to be pumped to the organs, esp the heart and brain. By shocking -using an AED or a cardiac defibrillator the heart can be shocked into a normal sinus rhythm. This is what is going on when you see those shows where they say "all clear, shock.".