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There are some Monsters that have special abilities, let's look at the Aboleth for example
Mucus Cloud (Ex)
While underwater, an aboleth exudes a cloud of transparent slime. All creatures adjacent to an aboleth must succeed on a DC 20 Fortitude save each round or lose the ability to breathe air (but gain the ability to breathe water) for 3 hours. Renewed contact with an aboleth's mucus cloud and failing another save extends the effect for another 3 hours. The save DC is Constitution-based.
My question is, that the affected Player Character has to roll Fortitude as the whole sum, or just roll Constitution with the Modifiers that player has
or that means that the monster has calculated the DC because of Modifiers, CR and such?
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When it says "Constitution based" or "Charisma Based" that is means the save DC of the monster is 10+Con or Charisma+ 1/2 its hit die + any racial modifiers. It does not mean the victim used his constitution or charisma specifically for the save. As far as i know without feats or home rules you never make a save with another ability score other than con for fortitude, wisdom for will, and dex for reflex.
To clear it up:
"While underwater, an aboleth exudes a cloud of transparent slime. All creatures adjacent to an aboleth must succeed on a DC 20 Fortitude save each round or lose the ability to breathe air (but gain the ability to breathe water) for 3 hours."
The bolded bit is all you need to know as far as how the affected victims must make their save. All the normal rules for Saving Throws (in this case it's Fortitude but not all abilities apply to Fortitude saves) are used and the monster's ability does not change the general rules for Saving Throws. If a monster has some ability that DOES change the general rules, that change will be explicitly described.
Note that the above bolded part describes the Saving Throw.
"Renewed contact with an aboleth's mucus cloud and failing another save extends the effect for another 3 hours. The save DC is Constitution-based."
This italicized bit describes only the DC of the Saving Throw which is, in this case, based on the aboleth's Constitution, as Falcar described.
Normally, you don't even need to care about this part; if a PC gets adjacent to an aboleth, tell them to roll their Fortitude Save and if they get a 20 or higher, they're OK, if they roll less than a 20 then they're affected. Period. All you need to know.
But...
Suppose one of the PCs uses a spell to lower the aboleth's Constitution by 4 points. This will lower the CON modifier by 2 points, which means that since the DC of the Saving Throw is based on this new, lower CON modifier, the Save DC just went from 20 to 18.
TL;dr:
So the only time you need to worry about the ability score used to calculate the Save DC of a monster's ability is when that ability score changes (better or worse) and you need to recalculate the new Save DC.