Executioner
The Executioner controls the "execution" behavior of the simulation. MOOSE includes several Executioners but most simulations will use the Transient Executioner, which is for simulating that is based on a PDE that has a time dependence (changes over time).
Grid Sequencing
Grid sequencing is a method particularly suited for nonlinear problems. It seeks to create an initial guess that is significantly close to the solution that Newton's method starts in or quickly enters the quadratic basin of convergence. The initial guess is generated by solving the finite element problem on one or a series of coarse grids and then prolonging the coarse grid solution to form the initial guess for the final fine grid. Grid sequencing can be conducted by simply specifying num_grids = n where n is the desired number of grids to perform the nonlinear solve on. In particular if num_grids = 2, there would be one coarse grid solve and then a fine grid solve, with the "fine grid" having one level of uniform refinement relative to the coarse grid. Note that in order to use grid sequencing, the initial mesh should correspond to the coarsest level of mesh desired because adaptivity in libMesh is incapable of coarsening a mesh beyond that initially provided by the user.
Attribute Reporting
The Executioner has a shortcut method that can be used to add an attribute reporting mechanism to the simulation. The Executioner accomplishes this by adding a special Postprocessors to the simulation that will report a potentially changing scalar value over time.