I worked as a phlebotomist for many years. It is fairly easy to tell if blood is taken from an artery instead of a vein. First of all, blood that comes from arteries is oxygenated so it is bright red in color. Blood that comes from veins is a deeper, darker red. In addition, arterial blood will pulsate (or spurt) into the blood collection tube. Blood from a vein will flow evenly into the tube.
Keep in mind that when drawing blood from a patient it is important to know the proper sites to draw from. Accidentally drawing blood from an artery can be dangerous. If blood is accidentally drawn from an artery, it is important to hold pressure on the site for several minutes to ensure that a hematoma will not form. Pressure needs to be applied from venous draws as well but not for as long.
The ideal spot to draw blood from is the median cubital vein which is in the bend of the arm. The tourniquet should be placed around the upper arm, tight enough to make the vein bulge. Some people have deeper veins and they may not be able to be seen. Veins have a very "springy" feel to them. The needle should be inserted bevel side up or the patient will feel added discomfort.
There are many specific steps that should be taken when drawing blood from a patient. I have listed a website below that goes into detail regarding the process.