2-morphism
A 2-morphism in an n-category is a k-morphism for $k = 2$: it is a higher morphism between ordinary 1-morphisms.
So in the hierarchy of $n$-categories, the first step where 2-morphisms appear is in a 2-category. This includes cases such as bicategory, 2-groupoid or double category.
There are different geometric shapes for higher structures: globes, simplices, cubes, etc. Accordingly, 2-morphisms may appear in different guises:
A globular $2$-morphism looks like this:
A simplicial $2$-morphism looks like this:
A cubical $2$-morphism looks like this:
Of course, using identity morphisms and composition, we can turn one into the other; which is more fundamental depends on which shapes you prefer.
Eric: Are there any consistency requirements for a 2-morphism? For example, in the bigon above, if $f:a\to b$, $g:a\to b$, and $\alpha:f\to g$, are there requirements on $\alpha:f\to g$ regarding $f$ and $g$? For example, should $\alpha$ come with component 1-morphisms $\alpha_a:a\to a$ and $\alpha_b:b\to b$ such that
or maybe
? Could there be a 2-morphism without the corresponding 1-morphism components?
Urs Schreiber: in any given 2-category you have to specify which 2-morphisms exactly there are supposed to be, what $\alpha$ exactly you allow between $f$ and $g$. When you ask about components, it seems you are thinking of 2-morphisms specifically in the 2-category Cat. Here, yes, the allowed 2-morphisms are those that are natural transformations between their source and target 1-morphisms, which are functors.
Eric: I think the exchange law might be what I had in mind.
In the 2-category Cat, 2-morphisms are natural transformations between functors.
In a path 2-groupoid 2-morphisms are certain surfaces or images of surfaces in a space, going between paths in that space.