Let's take a patient whose blood type is AB+. (If you don't know what "AB+" means, look at this post first.) These patients won't have any antibodies in their systems against A, B, or the rhesus groups. Thus they are the "universal receivers" referred to.
Conversely, patients who are group O- will have antibodies against the A and B red cell antigens, and possibly rhesus too. Thus, allowing them to receive anything but O- blood is very likely to end up causing a transfusion reaction. However, their blood can be given to just about anyone - since it lacks A, B and rhesus antigens it will therefore not be a target of any of these antibodies. Thus, O- people are the "universal donors".
HOWEVER, quips like "universal donor" are grossly misleading. Remember that there are hundreds of red cell antigens to consider. The ABO and rhesus systems are simply the most likely to generate a vigorous immune response if they are not matched - but lots of other antigens can do the same. This is why blood banks always match a blood sample from the patient needing a transfusion against the blood which is going to be transfused. Even when the ABO and rhesus systems match, there is often a mini transfusion reaction in the blood bank's test tubes, and they have to try other blood instead.
The only time you would be happy giving O- blood to a patient is when they need blood now or they'll die. The risk of a tranfusion reaction with this "universal donor" blood is then outweighed by the risk of not transfusing the patient there and then. If the patient can wait, a full crossmatch can often be performed within an hour or two - and this is much safer.