Although all animators have different techniques and styles. Its undoubted that they use similar sets of tools to do their work in 3D applications. The most popular for deforming a character/model would be latices and bones. All the other modifiers are also useful but none so as these two.
First: The Lattice.
Consider your objects in 3D as a kind of wire mesh that can be manipulated at the very points that one piece of wire joins another. This is known as a vertex.
In blender you choose this level by selecting edit mode and then the vertex select icon. The next level up is edge mode - The wire itself. In blender you choose this level by selecting edit mode and then the edge select icon. Finally the level before object mode is face mode. In blender you choose this level by selecting edit mode and then the face select icon. It is important to note that at the vertex level you can group and name the group. Once a group is created modifiers can be added that will allow the animator to manipulate the character/model while setting key frames. This is how an animator can achieve minute detail in an expression.
In this tutorial the tutor is selecting a collection of vertex, grouping them and then applying a lattice modifier. A lattice is like an external force field that will push and pull your model in a multitude of directions. Once applied to groupings of vertex you can use it to pull only small parts of your model. This is great for animating mouths and eyelids.
By far the most utilized deformer is the bone. Bones are of course internal to your model and deform your model within a given area. This is much like a force field emanating from the bone itself. You can use a bone in a similar way to the lattice but as a rule the bone has a parent child relationship to other bones. This means that one bone has control over another. Much the same way your arm has control of where your hand is and your hand has control over your fingers.
( Arm-Parents > Hand-Parents > Fingers )
It is by this process that you can build very complex animatable models.