Why is allyl chloride more reactive towards substitution than alkyl chloride?
But how allyl halide is more reactive than alkyl halide?

Reactivity towards $\ce{S_N1}$
The key to reactivity towards $\ce{S_N1}$ is the stability of the formed carbocation. Allyl system stabilises the carbocation through overlap with the vacant p orbital (@gsurfer999 has shown the resonance structures in his answer below). However, note that any allyl halide wouldn't be better at $\ce{S_N1}$ than any alkyl chloride.
In fact, the tertiary alkyl chloride's rate of $\ce{S_N1}$ is faster than that of allylic chloride which is secondary at one end. (due to $\ce{+I}$ effect of 3 methyl groups as well as hyperconjugation)
Visual aid for stabilisation through hyperconugation (from Clayden):

For such comparison, here's a useful table again from Clayden: 
Reactivity towards $\ce{S_N2}$:
In $\ce{S_N2}$ the most important factor is the substrate. The reaction proceeds through a single transition state with trigonal bipyramidal geometry and $\ce{\sim sp^2}$ hybridisation of the electrophilic carbon. The p-orbital makes two partial bonds, one with the nucleophile and another with the leaving group. Thus, it's electron deficient.  In such a case, the allyl system provides the additional electron density through conjugation.
However, again note that primary alkyl chloride is better at $\ce{S_N2}$ than allylic system due to least steric hindrance!