How are forces related to decays? How are decays related to forces, what is meant by particle X decays through the, say, strong force?    
The way I understand forces is by how they change the acceleration of particles with the right charge (mass, electric etc), through F=ma, how does it cause one particle to turn into other?    
How is it determined which force is responsible for which decays?  
Can a particle decay through the gravitational interaction?
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How are decays related to forces, what is meant by particle X decays through the, say, strong force?

It means that particle X was bound by the strong force to particle Y to form particle Z for a delta(t) and then it parts company.

The way I understand forces is by how they change the acceleration of particles with the right charge (mass, electric etc), through F=ma, how does it cause one particle to turn into other?

Your understanding is about the classical domain of forces.
Elementary particles necessarily belong to the quantum domain, one uses relativistic four vectors to describe them and forces are mediated by other elementary particles, graphically shown with Feynman diagrams, which are a short hand for the way one can calculate the probability of the decay. One can think of the mediating particle as a carrier of delta(p), where p is the four momentum, which transfers momentum and energy to the decay products from the binding energy of the original particle.

From a teacher site here is the Feynman diagram of the decay of a lambda baryon to a proton and a pion, mediated by the weak interaction:

Lambda contains a strange quark which is not stable under weak interactions. For a time the three quarks are bound together by the strong force, with gluon exchanges ( Not shown) but then the s quark decays into an up quark which is bound into a proton, and a W- weak boson off mass shell goes into an anti up and a down quark making a pi- .

How is it determined which force is responsible for which decays?

It has been determined experimentally and theoretically encoded into the Standard Model

Can a particle decay through the gravitational interaction?

We have not found elementary particles or resonances bound by the gravitational interaction so the answer is that experimentally it cannot. The reason is that the gravitational force is many orders of magnitude weaker than the other three forces that reign in the particle domain.
