Family of "something very close to be a curve" over a curve $C$ Hartshorne (IMHO restrictive) definition of a curve:
Definition of (complex) curve: A curve is an integral separated scheme of finite type over $\mathbb C$ of dimension $1$. (The definition of a surface is the same but clearly the dimension is $2$)
Most common definition of a family of curves:
Definition of family of curves:
If $C$ is a non-singular projective curve, a family of curves over $C$ is a non-singular projective surface $S$ with a flat morphism $f:S\longrightarrow C$. 
Now for $y\in C$ (closed point), consider the fiber  $S_y:=S\times_C\text{ Spec } \mathbb{C}$: it is a scheme of dimension $1$ and of finite type over $\mathbb C$, but in gneral it is not a curve in the sense of the above definition.
It is evident that the objects "curve" and "family of curves" are in contradiction each other since $X$ in a parameterized family of object that could not be curves. 
Can you suggest an idea to solve this problem? Maybe the most straightforward solution is to allow also non-integral curves.
 A: The moral is clear: even good families sometimes contain bad individuals. 
More seriously, this is just a matter of context: what kind of objects do we want to talk about right now?
Hartshorne's definition of curve you mention is part of his definition of variety, in Chapter II. I guess he includes "integral" because he wants to talk about the function field of a variety (and also because this goes along with the historically prevalent definition of variety).
If you go further in Hartshorne, at the start of Chapter IV he redefines curve: now it must also be nonsingular at every point. Why the change? Not because he changed his mind about what curve really means, just because he wants to discuss Riemann–Roch, Riemann–Hurwitz and so on, and doesn't want to write "nonsingular" fifty times. (And nor do we want to read it.)
In the other direction, a morphism $f: S \rightarrow C$ from a surface to a curve will often have some fibres that are reducible or non-reduced, but "family of curves" is a much more concise, user-friendly term for such a morphism than "family of connected pure 1-dimensional proper schemes". 
So it is really a question of convenience, rather than any philosophical dispute over what curve really means. One potential downside of this pragmatic approach is that the reader of a book or paper must be somewhat vigilant to ensure that they know exactly what the author means by curve in any given statement. But that is the (small) price you pay for well-written, concise mathematics.
