If you assume that your category $\mathcal C$ has enough projectives, then your projective arrows are the maps which factor through a projective. Let us check this.

Let $f\colon X\rightarrow Y$ be a projective arrow. Since $\mathcal C$ has enough projectives, we can take an epimorphism $g\colon P\twoheadrightarrow Y$ with projective source, hence $f$ factors through $P$. Conversely, if $f\colon X\rightarrow Y$ factors as

$$f\colon X\stackrel{f''}\rightarrow Q\stackrel{f'}\rightarrow Y$$

with $Q$ projective, then $f'$ can be lifted along any epimorphism $g\colon P\twoheadrightarrow Y$, hence $f$ too.

Maps which factor through epimorphisms are widely studies in homological algebra. The stable category $\underline{\mathcal A}$ of an abelian category $\mathcal A$ is the quotient of $\mathcal A$ by the ideal of morphisms factoring through a projective, i.e. by the ideal of projective maps in your terminology. You can read in wikipedia about the stable category of modules over a ring:

http://en.wikipedia.org/wiki/Stable_module_category

if ${\mathcal A}$ is a Frobenius abelian category, i.e. enough projectives and injectives and both classes of objects coincide, then $\underline{\mathcal A}$ is a triangulated category. All algebraic triangulated categories arise in this way, but allowing ${\mathcal A}$ to be an exact category, not just abelian.

If you assume that your category $\mathcal C$ has enough projectives, e.g. the category of groups, then your projective arrows are the maps which factor through a projective. Let us check this.

Let $f\colon X\rightarrow Y$ be a projective arrow. Since $\mathcal C$ has enough projectives, we can take an epimorphism $g\colon P\twoheadrightarrow Y$ with projective source, hence $f$ factors through $P$. Conversely, if $f\colon X\rightarrow Y$ factors as

$$f\colon X\stackrel{f''}\rightarrow Q\stackrel{f'}\rightarrow Y$$

with $Q$ projective, then $f'$ can be lifted along any epimorphism $g\colon P\twoheadrightarrow Y$, hence $f$ too.

Maps which factor through a projective are widely studied in homological algebra. The stable category $\underline{\mathcal A}$ of an abelian category $\mathcal A$ is the quotient of $\mathcal A$ by the ideal of morphisms factoring through a projective, i.e. by the ideal of projective maps in your terminology. You can read in wikipedia about the stable category of modules over a ring:

http://en.wikipedia.org/wiki/Stable_module_category

if ${\mathcal A}$ is a Frobenius abelian category, i.e. enough projectives and injectives and both classes of objects coincide, then $\underline{\mathcal A}$ is a triangulated category. All algebraic triangulated categories arise in this way, but allowing ${\mathcal A}$ to be an exact category, not just abelian.