Complete cover: No hide check is needed, though one is a good idea as cover has a way of being negated. As long as there is complete cover, you and the foe are unable to interact with line of sight or line of effect based rules. As your foe (or you or the cover) move, the cover may go from total cover to a lesser degree of cover or may be completely removed. If you are 'hiding' with complete cover behind a boulder and the enemy wizard disintegrates it, you are no longer hidden.
Partial cover or concealment: When you have cover (less than total) or concealment between you and your foe, you may make a hide check. If successful, the foe will not notice you. If that cover or concelament stops blocking the line of effect/view between you and the observer, your hiding ends. So, if you're hiding behind a rock and then leap out to attack a wandering kobold 10' away, you are no longer hiding when you reach the kobold. Of course, the kobold will probably be flat footed ...
You need to ask your DM whether an obstruction provides cover or concealment in questionable cases. If a rogue or monk slips behind some thin bushes, do they provide cover or concealment? That is up to the DM ... and may change depending upon why you're worried about the cover/concealment. I might treat some thin bushes as cover for purposes of a ray attack (disintegrate, ray of frost, etc ...), but as concealment for purposes of a physical weapon attack (arrows). For purposes of hiding, it is generally better to error on the side of concealment (for reasons explained below).
Let us say the monk is hiding behind a boulder that provides complete cover against an enemy fighter. That enemy fighter then walks past the boulder. At one point during his path, that fighter will go from complete cover to partial cover. At this point, you will begin to need to worry about the monk's hide skill. As the fighter continues, that partial cover will become no cover. Unless the monk has 'hide in plain sight' or camouflage abilities, he will be seen, regardless of a hide roll.
Look at your number keypad on your keyboard. It should be laid out like this:
7 8 9
4 5 6
1 2 3
Imagine 7, 4 and 5 are occupid by a large boulder. 8 is the location of a monk. An enemy fighter is walking along a path from 1 to 2 to 3 to 6 to 9. While in spaces 1 and 2, the fighter and monk each have complete cover. As he moves into area 3, there is no longer complete cover. There is line of effect between the monk and the fighter (upper right corner of squares 3 and 8 can be connected without crossing a barrier), so we now need to consider the hide rules. The DM may consider the cover to be sufficient to qualify for the varying degrees of cover rules, but likely not, so if the monk is hiding, he gets a hide check opposed by the fighter's spot to remain hidden.
Assuming the monk manages to make the hide check, the fighter then moves to square 6. The partial cover remains and the smae hide check is in effect. Area 6 is within the reach of the monk, so if he had a readied attack, he could make that readied attack, though the fighter would get an AC bonus for cover (+4).
The fighter then moves to space 9. The monk is threatening square 6, so as the fighter moves out of square 6 we must consider whether he provokes an AoO for moving out of a threatened square. The rules for cover state that you may not make an AoO against a foe with cover. The AoO is created for attacking a foe as they leave a threatened square, not for entering a threatened square. Accordingly, as he leaves the square, he still has cover from being in square 6, so he does provoke an AoO, but the cover rules do not let you take that AoO against the fighter.
After moving into square 9, there is no cover between the monk and the fighter. Unless the monk has a hide in plain sight ability, he is seen regardless of his hide role.
If the monk wants to stay hidden and make that AoO in a situation such as this, he needs to use concealment instead of cover to block the view. Concelament does not block AoOs like cover does. In most circumstances, the monk will need to consider his hide check earlier as the view through the concealment will often be possible earlier than if you have solid and complete cover.
As a final note: Some people point to word 'attempt' in the hide description ("You need cover or concealment in order to attempt a Hide check.") and indicate that cover or concealment is needed only when hiding begins - when the attempt is made. This is a possible interpretation under the rules as written, but it is illogical in the presence of abilities like camoflage and hide in plain sight that would become somewhat redundant, making only a minor difference to a high level character (where these abilities are gained) as opposed to being powerful abilities for powerful PCs. 'Attempt' is more likely to be there to reference the fact that it is possible to fail the hide check.
