Proficiency in a skill is used when... you are proficient in that skill. The exact amount changes by level. See the chart in the game rules, but it starts at +2 and eventually goes to +6.
From your example, the fighter (with a wisdom of 12 or 13, for a +1 wisdom modifier) has a perception modifier of +3, so that example character must be trained in perception. Perception is on the fighter's list for potential proficiencies (choose two from a list), and clearly the example character did choose that (or got it from a background choice).
Passive perception is not 14, because the passive score is just 10 + your regular bonus for that skill, not 10 + your ability modifier + your regular skill bonus which already includes your ability modifier.
The dwarven cleric isn't trained in perception, but has a high wisdom, so the +3 wisdom modfier = the skill modifier too (no proficiency bonus is added) and the resulting passive score is 10 + just the ability modifer — so, also 13, but for different reasons.
In general, the proficiency modifier is used when you are trained in (or, in other words are proficient with) the skill, tool, or weapon in question.
This is explained in Chapter 7 of the freely-downloadable basic rules, and the sidebar about Hiding on page 60 explains all of this pretty nicely. In the official terminology, there does not appear to be a name for the number you write next to the skill, other than the relevant ability modifier plus bonuses and penalties. But, colloquially, I think many people will use the 3rd edition terminology and call this a "skill modifier" or "skill bonus", or specifically "perception bonus". People will say "perception skill" too, although I think that should be avoided because it could arguably mean either the bonus, the passive score, an actual roll + modifier, or even the just whether you're trained or not.