Why is allyl chloride more reactive towards substitution than alkyl chloride? I can understand that the reactivity of alkyl halide is more than vinyl and aryl halides because the halogen atom in alkyl halide is connected to the sp3 carbon. 

Whereas in vinyl or aryl the halogen atom is connected to the sp2 carbon. 

But how allyl halide is more reactive than alkyl halide?
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!