The definition of a simplicial object is a functor $X_\bullet\colon \Delta\to Delta^{op}\to A$ where $\Delta$ is the simplicial category and $A$ is your favorite category. So the easiest answer is that a simplicial object in a category is a sequence of objects in that category together with morphisms $d_i$ and $s_j$ that satisfy a bunch of relations.
We can also think of a simplicial object $X_\bullet$ as an element of $Fun(\Delta, Fun(\Delta^{op}, A)$, the category of functors from $\Delta$ \Delta^{op}$ to $A$. In this context, it is easy to define a morphism between simplicial objects. It maps $X_n\to Y_n$ and commutes with the $d_i$'s and $s_j$'s.
Shameless planar algebra plug: Simplicial objects are also really cool in tandem with an adjoint functor pair. You can use this machinery to get a pictorial representation of the simplicial category using Temperley-Lieb (string) diagrams. In fact, planar algebras are great examples of simplicial vector spaces, although there's a lot more structure too...
