In an $n$-category, or most generally an $\infty$-category, there are many levels of morphism, parametrised by natural numbers. Those at level $k$ are called $k$-morphisms or $k$-cells.
a 0-morphism is an object
a 1-morphism is a morphism
next are 2-morphisms
and so on
All notions of higher category have $k$-morphisms, but the shapes may depend on the model (or theory) employed.
For a simplicially based geometric model of higher categories, i.e., simplicial sets subject to some filler conditions, the $k$-morphisms are literally $k$-cells in the sense of a simplicial set. This applies for example to quasi-categories, weak $n$-categories in the sense of Street, and the weak complicial sets of Verity. In other geometric models, based not on simplices but on other shapes such as opetopes (Baez-Dolan), multitopes (Hermida-Makkai-Power), or $n$-disks (Joyal), a higher category is a presheaf
again subject to some filler conditions, and in each case $k$-morphisms are elements of $X(\sigma)$ where $\sigma$ is a shape of dimension $k$. Still other shapes (e.g., cubes) are possible (see also n-fold category).
Many notions of algebraic higher category, such as those due to Batanin, Leinster, Penon, and Trimble, are algebras over certain monads acting on globular sets (such as those induced by globular operads), so that each higher category $X$ has an underlying globular set $U(X)$. In that case, the $k$-morphisms are the $k$-cells of $U(X)$. In such globularly based definitions, every $k$-morphism $f$ has a $(k-1)$-morphism $\sigma f$ as its source and a $(k-1)$-morphism $\tau f$ as its target, and the source $(k-2)$-morphisms $\sigma \sigma f$ and $\sigma \tau f$ must be the same, as must the target $(k-2)$-morphisms $\tau \sigma f$ and $\tau \tau f$.
A $1$-morphism may simply be called a morphism; a $0$-morphism is an object.
For the purposes of negative thinking, it may be useful to recognise that every $\infty$-category has a $(-1)$-morphism, which is the source and target of every object. (In the geometric picture, this comes as the $(-1)$-simplex of an augmented simplicial set.)
every (non-empty? -David R) $\infty$-category
I think every. Up to equivalence, a $k$-morphism in $C$ is given by a functor from the oriented $k$-simplex to $C$. As the $(-1)$-simplex is empty, there is a unique such functor for every $C$; thus every $C$ has a unique $(-1)$-morphism.
Also note that every $k$-morphism has $k + 1$ identity $(k+1)$-morphisms, which just happen to all be the same (which can be made a result of the Eckmann–Hilton argument). Thus, the $(-1)$-morphism has $0$ identity $0$-morphisms, so we don't need any object. (This confused me once.)
k-morphism