MooseMesh
MooseMesh is the base class that all mesh objects in MOOSE must inherit from. MooseMesh is responsible for holding the underlying data structures holding the actual mesh (through libMesh) and caches for many commonly accessed entities. MOOSE currently requires an active Mesh for every simulation even if the mesh is not strictly necessary when wrapping a third party application or running non-FE type calculations. The Mesh however can consist of just a single element.
Side sets and Node sets
MOOSE can apply several different objects to side sets and node sets to either compute quantities of interest or set boundary conditions. It is important to understand the different between these two concepts. Node sets are simply a set of nodes, typically on a boundary, that all contain a common ID (the node set ID). These can be created and assigned in your mesh builder program (such as Cubit or GMsh). Side sets are a collection of elements (sides or edges), typically on a boundary or on a plane within your mesh. These can be created and assigned in your mesh building program.
By default, MOOSE will construct a node set corresponding to each side set within your mesh. This means that if you always prefer side sets (to node sets) you won't have any issues applying any kind of boundary condition or other "boundary" type object within MOOSE.
More detail
As users of MOOSE, you do have the ability to control whether side sets or node sets are automatically constructed. By default node sets are constructed from side sets but the converse is not true. Both of these parameters can be controlled by the following parameters (respectively)
Mesh/construct_node_list_from_side_list=true
Mesh/construct_side_list_from_node_list=false
If you have a mesh that is missing a side set (but it has a node set) you may be able to get around that issue by setting the second parameter (from above) to true. The construction will happen at the end of the mesh generation process.
You could also use the SideSetsFromNodeSetsGenerator class to construct side sets from node sets (or the NodeSetsFromSideSetsGenerator for vice versa). These object allows you to manually construct the relevant side sets and node sets earlier in the mesh generation process.
To declare subdomain IDs which are not found on the mesh right after the mesh is setup, a list of IDs can be specified using the Mesh/add_subdomain_ids parameter. This permits setting up block restricted objects that will only act on subdomains created during the simulation (e.g. using the CoupledVarThresholdElementSubdomainModifier).
Minimal interface
When inheriting from MooseMesh, the developer is responsible for creating the mesh and cloning the mesh (used for displaced mesh problems). The methods that need to be overridden are "buildMesh" and "safeClone".