Good questions. I'll explain the different abbreviations you might see. We Muslims never say a Prophets name just as it is...
(PBUH) - This stands for "Peace be upon him".
(SAW) - This is arabic for "Sallalahu Aleyhi Wasallam" which means "May the Peace and Blessings of Allah (God) be upon him"
(AS) - This is arabic for "Alayhi Salaam" which means "upon him be peace".
The 1st and 3rd are given to Prophets of Allah but the 2nd one is recited after the Holy Prophet (SAW)'s name.