How does the saturation flux density affect magnetic field strength?

[yosimba2000]

So I'm confused what the Saturation Flux Density is referring to. definition says it is when you no longer get an increase in H-field when increasing external B-field.

So, does the satuation flux mean the core can only create fields UP TO that saturation flux, or that it can make a stronger field but accept at most the saturation flux?

For example, let's say I have a 1.5T magnet and a core with 1.5T saturation flux. When I pair the two, will the core give me a magnet STRONGER than 1.5T (say 2T), or will it just give me a larger 1.5T magnet?

 

[anuttarasammyak]

H-field comes from spin direction of the material. When all the spins have same direction they show saturation flux density.