Analgesia means the loss of ability to feel pain while conscious, the state you will be in during Conscious Sedation.
There is a significant difference between Intravenous/IV Conscious Sedation and general anesthesia (unconscious sedation) used for invasive surgical procedures. During general anesthesia, you are completely asleep and cannot be aroused — not even by painful stimulation, and it requires breathing support, plus an anesthesiologist is needed to monitor your vital signs. During IV Conscious sedation you are not “asleep” but rather in “Twilight Sleep,” a much safer and more comfortable semi-awake state of reduced anxiety, but you are still able to follow the dentist’s instructions, like “open wide.” However, the feeling of relaxation is so profound that you are unlikely to remember much about the procedure. As a result, time will appear to pass very quickly, and you will be out of the chair before you know it, the whole time free from anxiety, and without a memory of the procedure to disturb your dreams.