akirusan basically got it right.
Your conjurer spells increase affinity when cross-classed.
For example, lets say you heal 200 on Lancer. With One with Nature, you'll heal something like 210. Cure will still be single target.
And for the tranquility example, you will gain 5% of your mana back on a 20 minute cooldown (So if you have 500 MP as a lancer, you will gain 25 back).
That being said, it's not the same for all skills though. With lancer's Pikemanship, I cannot cast Speed / Life surge on my conjurer.
And I have an inkling that more affinity skills will be purchaseable when you get to higher rank (R60 R70). Right now you can get a maximum of 10 (You cannot buy the skill multiple times). However, since it says for a maximum of 100, that suggests that at the higher ranks, you will be able to buy things like One with Nature II / III /Pikemanship II/ III/ etc.