What do all the common maps "preserve"? I am trying to organize all the different "morphisms" in my head. To help me remember, I am trying to imagine what these maps "preserve." By "preserve" I mean an invariant under the mapping. Can you help me fill in the gaps?
An isometry of a manifold preserves lengths and angles.
An isometry of a vector space preserves the inner product.
A homeomorphism of a manifold preserves topological properties (how do we quantify this?)
A diffeomorphism of a manifold preserves "smoothness" as well as topological properties.
An injection between any two sets preserves "uniqueness"
A surjection between any two sets preserves the "wholeness" of a set
An isomorphism between vector spaces preserves operations: scalar multiplication and vector addition.
A morphism preserves...?
Is this a good way of categorizing these maps? Should I be remembering them differently?

Though your examples might require some refinements (as written some in the comments), you basically got the idea: there is something common in whatever kind of 'morphisms'.
And the most general thing in common, is that idenitites will always be included and they all are closed under composition, which makes their class a category.
Most of your lines specifically take the invertable ones of the given type of morphisms. (E.g. homeomorphisms are the invertible morphisms in the category 
$\quad$ [objects: topological spaces, morphisms: continuous functions].)
