If your course started with an indigestible dose of multi-dimensional calculus without much discussion of how the math relates to the physics (and there are certainly textbooks on computational CFD which start that way!) the answers to these questions may become clearer when you start developing the actual numerical methods and seeing how they perform solving "real" problems.
There are two basic ways to devise a numerical solution scheme. One way is to consider the values of quantities defined at specific points in space (e.g. the grid points of a finite difference mesh). The computational method ignores what happens to the quantities "in between" the mesh points, and you hope that as you increase the number of points, the computed values will converge to the solution of the underlying differential equations. (Sometimes, you can prove the solution will converge in this sense, but for most "real world" problems involving nonlinear differential equations you can't prove it).
The other way is to consider the value of properties over regions of space - for example the mass, energy and momentum of a "unit cell" defined by the grid points that surround it, and the flows into and out of the cell across its boundaries. 432
This has the advantage that the solution method can guarantee that it satisfies some fundamental physical laws like conservation of mass and energy. Usually, a solution method based only on values at specific points will not automatically satisfy such conservation laws, though (if it is a useful solution method!) increasing the number of solution points should reduce the errors.
But satisfying the global conservation laws doesn't necessarily mean that the grid-point values are "accurate" - for example there may be unphysical oscillations at adjacent grid points which don't affect the "average" energy, momentum, etc.
Also our lecturer mentioned that the Jacobian strong form is in Lagrangian form but differential is in Eulerian form. I know the meaning of both but I am not sure what it means in this context.
Without knowing exactly what was said (or in the lecture notes) I'm not sure what that means either. You can certainly create four versions of the mathematical formulation, i.e. weak and strong forms of both the Lagrangian or Euleran equations - though some are more intuitively "obvious" than others. In fact some numerical methods use a combination of more than one of the four options. 499