Are you interested in the physics or the chemistry? The physics has several theories for the explanation of atom's and how they interact with one another. The current favored flavor is Density Functional Theory if I recall correctly(something I know nothing about). Many Physical Chemists use electrolysis for experiments, so Physical Chemistry is the realm you are looking for(I think?).
I only have a surface level understanding of such things. I can be a reductionist and try and answer some of these
Well you ask what bonds two atom's together. To be very brief, electrons do. Notice the key-fragment "Electr"
. There are many kinds of atomic bonds. In general we have Covalent and Ionic(some bonds fall into a grey area between the two). Covalent(sigma and pi bonds) for organic molecules and Ionic for ionic compounds(very strong ionic bonds).
How does Electricity disturb or break these bonds? Well, I can't tell you exactly how, not sure if anyone
really can. This little article may lend a tiny but more insight,
http://www.ibchem.com/faq/?p=8. In general we look at oxidation/redox reactions as depicted here,
http://chemed.chem.purdu...bp/ch20/electro.php#half .
Does this involve a Magnetic Field? Not really sure, but here's an experiment that might lend some insight at least situationally
http://pubs.acs.org/doi/abs/10.1021/jp010581u .
What you are describing in general with the two electrolye containers is called a half-cell. Where two containers are seperated by a semi-permeable membrane such as a salt-bridge. The reason for the two containers being separated is a thermodynamics one. By separating the two cell's we generate more energy by form of Work.
Sometimes experiments are done in what are called 'earthen-ware' containers in older literature, which to my understanding is just a semi-porous flower-pot like container.
I know a bit more on this topic, FAR from an expert but it also highly interests me!