I had the dubious honour of making one myself not long ago (a Pan call, that is).
I had already informed ATC we had a problem which we were trying to resolve, so he was aware something might be in the offing. After a few minutes we decided we could not resolve the problem, and diverted. ATC asked if I was declaring an emergency, to which the reply was, "Affirm, Pan Pan," etc.
Normal callsign was used from then, and when I called the diversion airfield. On first contact with them, about 10 minutes after the Pan call, they informed us the emergency services were on the way, the adjacent road would be closed for our approach, and a neighboring airfield was standing by just in case we needed to go there. When we arrived, a short time later, there were dozens of professional people waiting, all with the sole purpose of helping us out of our difficulty.
The word "Pan" is absolutely magic. It performs miracles, as does "Mayday". Either word instantly removes any doubt or confusion regarding the seriousness of the situation. Any pilot who does not use the word when the situation demands is an idiot.