Clerics and Druids receive 8-sided dice for hit point progression (see Player's Handbook, Chapter Three). The cleric does a lot of fighting, and the druid has a very demanding existence, living as he does in the wilderness; both need to have comparatively high hit point totals.
The DM can, if he wishes, make any priesthood of a specific mythos take six-sided dice for hit points (like rogues) or even four-sided dice (like wizards). But this is only appropriate for priestly orders which are not very demanding physically.
Most priesthoods should have eight-sided dice for hit points. If a DM decides that a priesthood will have less robust dice, then he must compensate the priesthood with enhanced access to spells and (especially) numerous Granted Powers (discussed below).
A priesthood should only receive six-sided dice if it has medium to poor combat abilities, and belongs to a deity with very few physical demands.
For example, a god of dawn has no intrinsic orientation toward combat, and "dawn" does not imply any specific physical demands.
A priesthood should only receive four-sided dice if it has poor combat abilities, and belongs to a deity whose attribute implies soft living.
For example, a god of peace or love could have priests with little or no combat abilities and with four-sided dice. However, it's important to note that this does not have to be the case... It is only the case when the DM insists upon it. Priests of the god of peace could be pacifists, but still be good at combat... which they may only employ in self-defense. A god of love noted for weapons use, as Eros was noted for his archery, could have priests who specialized in that weapon, and would not have to be stuck with a measly 1d4 for hit points.
Magical Items[]
Priests are already limited to using magical items usable by all classes or by priests only. But they might also be further restricted by their faiths.
For example, priests of a specific faith might be required to use only magical items made by priests of their order, or might be required to use no magical items whatsoever.
Mutilation[]
Occasionally, a priesthood will demand a sacrifice of mutilation of its priesthood. This is most common among evil priesthoods, but could theoretically occur with any priesthood, regardless of alignment.
For example, devotees of a blind god of prophecy might be forced to blind themselves. (However, it would be more appropriate for them merely to have a Clothing restriction that required them to wear blindfolds when performing official duties.)
When this sort of thing does take place, the priesthood is often compensated for its loss: The god often grants them an extra power (as described below, under "Powers of the Priest"). A priesthood required to be blind, for example, might have an extra power of analysis, identification or prophecy.
Weapons[]
Many priesthoods require their priests to use only a certain category of weapons. Some only restrict them from using a certain narrow category of weapons. Some require their priests to use no weapons at all. Commonly, a god identified with a certain type of weapon will require his priests to use that weapon and a certain number of similar or related weapons.
The DM should assign the priesthood a weapons restriction based on two choices.
First, weapons restrictions reinforce the special attributes and character of the worshipped god; limiting priests of the god of Death to sickle-like weapons certainly establishes flavor for them.
Second, weapons restrictions, like armor restrictions, help define the combat roles of priests in the campaign. If a priest is limited to daggers and creampuffs, he's not going to be the campaign's combat monster, so magic and his priestly duties will be much more important to the character.
Many examples of this are given in the next chapter, "Sample Priesthoods."