
Lord Zordran |

The wording in the Advanced Player's Guide is a leftover from the playtest version of the Inquisitor, which had the bonuses from the Judgement class ability start at +1 and then increase each round up to a certain maximum. This was deemed to be too much bookkeeping to be worth the effort, so the the Judgement class ability now provides a fixed bonus based on the Inquisitor's level. The errata for the Slayer class ability reads:
Slayer (Ex): At 17th level, an inquisitor learns to focus her judgment. Whenever an inquisitor uses her judgment ability, she must select one type of judgment. She is treated as if she were 5 levels higher for the purposes of determining the bonus granted by the judgment. Unlike other types of judgment, the one enhanced by this ability cannot be changed for the remainder of the judgment.

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Are you talking about the one in the book, which allowed you to go straight to the '3rd round' effect'?
If so, it's a relic of the Beta test, when the bonuses grew during combat, and I believe it got eratta'ed to it's current form, available to view on PFSRD:
Hopefully this version is self-explanatory.
[Ninja'ed by Zordran!]

wraithstrike |

What is the function of this ability? I can't seem to find out what the purpose is... If you have the answer, please give an example.
"At 17th level, an inquisitor learns to focus her judgment. Whenever an inquisitor uses her judgment ability, she must select one type of judgment. She is treated as if she were 5 levels higher for the purposes of determining the bonus granted by this judgment. Unlike other types of judgment, the one enhanced by this ability cannot be changed for the remainder of the judgment."
The purpose of it is to boost a particular judgment of your choice so that it is more powerful than normal.
Let's say you choose destruction which is below. Normally at level 17 the bonus would be +5(17/3) but since you it would treated as though it were being used by a 22nd level inquisitor the bonus would be +7(22/3).
Destruction: The inquisitor is filled with divine wrath, gaining a +1 sacred bonus on all weapon damage rolls. This bonus increases by +1 for every three inquisitor levels she possesses.