Spin-dependent and spin-independent interactions of the dark matter What is meant by spin-dependent and spin-independent interactions of the dark matter with the nuclei?
How is the interaction between nucleus and DM modeled?
What can we conclude if the DM-nuclei interactions are spin-dependent and independent respectively?
How will the cross-section be modified in case of spin-dependent interaction?
I'm a beginner n dark matter theory and therefore, use of too many jargons will not be helpful.
 A: 
What is meant by spin-dependent and spin-independent interactions of the dark matter with the nuclei?

The nuclei and atoms  have spins, as do the free particles, protons electrons photons in the space environment.
This statement assumes an elementary particle not yet discovered is what constitutes dark matter, axions, or WIMPS. The interaction crossection will be different if the dark matter particle is an axion, which has spin 0 , or one of the WIMPS which could have  a spin different than zero.

PICASSO is predominantly sensitive to spin-dependent interactions of WIMPs with the fluorine atoms.

So spin dependence or independence is mathematically modeled and checked with data , in the various experiments.
A: At the typical speeds of WIMPs relative to the DM detector (of the order of the 300 km/s) the entire nucleus will fall well within a de Broglie wavelength. The amplitude for the scattering cross section is then the sum of the amplitudes of the interaction with all the nucleons separately. If the interactions are spin-independent, this means that the amplitude is proportional the number of nucleons.
However, in case of spin-dependent interaction, the amplitude changes sign if you reverse the spin of a nucleon. Then since the nucleons in a nucleus have alternating spins, the contribution to the total amplitude from different nucleons will cancel out, leaving you with a total that depending on the total spin can be of the same order as what is for scattering off a single nucleon or it can be almost zero.
The total cross section is proportional to the square of the amplitude, so this can be much larger for spin-independent interactions of WIMPS with heavy nuclei. 
