The Temple Knights are an elite section of the White Knights, dedicated to defeating any threat to Falador. To join, the applicant must be at least a White Knight. Sir Tiffy Cashien is head of recruitment. If you want to join, you must complete the Recruitment Drive quest.
It is reasonable to assume that they have been established for an extremely long time, as Juna the snake has heard of them, despite being in the Tears of Guthix cave since the end of the Third Age.
The Temple Knights origins can be traced back to the time of the God Wars. Saradomin himself founded the order, which was made up of his most devout human followers. It is said that these human followers also guarded the first temple of Saradomin.
The Temple Knights survived the God Wars, and in the Fifth Age their primary activity is protecting the kingdom of Asgarnia. They do so in secret, there being only rumours and fables of their works and actions, and there are no official records of their presence. They consider themselves the front line of defence for Asgarnia, being smarter and stronger than the White Knights. They are a very powerful organisation as they take their orders from, and answer only to, Saradomin.
Whether or not the White Knights are as "good" as they make out remains to be seen. Juna the snake, who is met during the Tears of Guthix quest, comments that their intentions may not be what they seem when the player tells her of their exploits involving the Temple Knights.
Currently there are two known ranks within the Temple Knight order: Initiate and Proselyte. Initates are White Knights newly inducted into the Temple Knight order and have access to Initiate armour while proselytes are basically promoted initiates and have access to both Initiate and Proselyte armour.