PHB descibes Jack of All Trades this way :
Starting at 2nd level you can add half your proficiency bonus to any ability check you make that does not already include your proficiency bonus.
it does not mention anything about untrained skills.
I did not know we could add proficiency bonus t our skills ... Could you possibly tell me how much bonus/points do we have in the skills that we choose, by default ? Lets say I get Persuasion and History through my Noble background, how much points or proficiency bonuses do I have there now at level 1 ?
Here's how it works: There are three types of success rolls in the game: ability checks, attack rolls, and saving throws. All three work similarly: roll 1d20, add the appropriate ability modifier, and (if you're proficient in that kind of activity) your proficiency bonus. Skill checks are a subset of ability checks - that's why they're always written as "roll Charisma (Persuasion)" rather than just "Persuasion".
If you have the skills Persuasion and History from your Noble background, you add your proficiency bonus to any Charisma (Persuasion) or Intelligence (History) check you make. So if you have Charisma 16 (bonus +3) and proficiency bonus +2 (for being 1st level), you would have a total of +5 to Charisma (Persuasion) checks. If you were instead trying to scare someone off you would roll Charisma (Intimidation), and since you're not proficient in Intimidation that would just add the +3 from Charisma.
Once you get Jack of All Trades, you get to add half your proficiency bonus to
any ability check that wouldn't normally add it. So it doesn't make you any better at Persuasion - you're already proficient in that. But you
do get to add half your proficiency bonus to that Intimidation check we mentioned earlier, so now that's at +4 instead of +3. This also applies to ability checks that normally don't fall under any skill at all, for example Initiative checks (which are Dexterity checks), or the the contested Intelligence check used for the
detect thoughts spell. However, it does
not apply to attack rolls or saving throws, because those are not ability checks.
At 3rd level, you also get Expertise, which lets you
double your proficiency bonus for two skills. Now, if you choose Persuasion as one of your expertise skills, you would add an
additional +2 (proficiency bonus) to Charisma (Persuasion) checks, for a total of +7. Now that's some smooth talking.