How does p53 signaling network affect cancer development? How is it that certain mutation in DNA are repaired effectively and others are not even with the involvement of p53 protein?
Generally speaking, p53 pathways are "off". Stress signals (spindle damage, hypoxia, rNTP depletion, ribosome biogenesis, etc.) activate mediators in a complex network that can result in a number of "outputs" (from apoptosis to inhibiting other pathways such as the IGF-1/mTor). The central, most important component to understand is the core regulation (i.e., p53 & MDM2).
One druggable example: "Manganese Superoxide Dismutase Is a p53-Regulated Gene That Switches Cancers between Early and Advanced Stages" http://cancerres.aacrjournals.org/content/71/21/6684.long
I.e. if you do a pubmed search for "p53 + cancer" you'll get about 53.000 publications. If you filter for reviews, you'll obtain about 7.000 results. I suggest reading a few reviews in good journals ("Nature Reviews Cancer" is one of my favorites) to get a better understanding (or a mere glimpse) of what's going on with p53 in cancer.