Pakistan and India have deep-rooted differences dating back to early 1900's. When the British invaded India they followed the policy of Divide-and-Rule. Muslims and Hindus had lived in harmony for ages. They were the two major communities in the United India. The British started to favor Hindus and ignore Muslims, they would not allow a single opportunity to create differences among the two communities pass by. The Hindus became greedy of more power and over time their attitude towards Muslims changed. The British later on hinted at planning to leave India for good. The transfer of power was an issue that worried the Muslims. They did not wanted Hindus to get the absolute power since that would give them a license to increase their brutalities towards Muslims. A freedom struggle was launched and Muslims were granted a separate homeland where Muslim-majority areas existed in India (resulting in East Pakistan- now Bangladesh- and West Pakistan - now Pakistan). The partition was unfair in many ways. It was biased in favor of India. Critical areas and areas contiguous to Pakistan were given away to India ignoring geographical realities.
At the time of partition there were also several autonomous states under the rule of princes (thus called Princely States) which were told to accede to either Pakistan or India. All but four states' fate was decided.
One of them (Hyderabad) decided to remain independent after its initial decision to accede to Pakistan was challenged by India on the pretext of geographical restrictions. India forcefully captured Hyderabad months after independence and before the Nizam of Hyderabad (the ruler) could make a petition before the UN Security Council.
The twin states of Junagadh and Manawadar were on the coast of India less than 300km from the then capital of Pakistan - Karachi. They decided to accede to Pakistan. The Indian Navy besieged the two states before the Pakistan Navy could reach them and in an operation that was publicized as a LIBERATION OPERATION, they forcefully took control of the two states months after independence. Pakistan made an official petition before the UNSC that is still pending. The issue is not raised much.
The Princely State of Jammu and Kashmir was sold by the British to a Sikh Ruler Raja Dahir. The state was largely Muslim. The people of the state wanted to accede to Pakistan while the ruler was in favor of India. Under severe public pressue he approached the Indian Government for a military action. They pursued him to sign an instrument of accession and invaded the state of Jammu and Kashmir through the areas of Ferozpur and Gurdaspur in 1948 (unfairly given to India since they were geographically contigious to Pakistan). This triggered the FIRST WAR BETWEEN THE TWO NATIONS. The place where the Indian Army was finally halted is the Line of Control that divides the state between the Pakistan held (Liberated Kashmir or Azad Kashmir as the Pak Govt calls it) and Indian Occupied regions. The Indian Army fearing their operation may backfire more, filed a petition in the UNSC. The UNSC ordered immediate ceasefire (LoC is also called Ceasefire Line) and a plebiscite to know the wishes of the population of the state. No such plebiscite has ever been held there by the Indian Government. The largely Muslim population that wanted to accede to Pakistan even several years after the UNSC orders is now finally tired. They now want an Independent Kashmir or a Kashmir that is no longer divided by an LoC.
India is also held widely responsible for causing an uprising against West Pakistan in East Pakistan. The general elections in 1970 in Pakistan resulted in Sheikh Mujib of East Pakistan gaining a majority. But the set-up was rigged and Z. A. Bhutto of West Pakistan was instated as the Prime Minister. This incited mass riots in East Pakistan and the fragile conditions of East Pakistan worsened further. India got hold of this opportunity and offered complete military assistance to East Pakistan to enable it to declare independence. The whole issue was severely mishandled by the administrative and military set-up in Pakistan and the Pakistan Forces had to face humiliation when 91,000 soldiers surrendered in East Pakistan.
Despite several peace talks the main hurdle between the resolution of issues and differences remain largely the minds of Indians polluted by a wave of media in the past that had them brainwashed against Pakistanis, the hardliners on both sides of borders and Indian extremists who have been notorious to create chaos and mayhem whenever its something involving Muslims or Pakistan.
The conditions are now improving and people are now becoming more aware of ground realities. They are now coming out of the biased history that's been fed to them and are now seeing things from a more unbiased persepective.
Wars fought between Pak and India:
-->1948 Kashmir War [ceasefire ordered shortly]
-->1965 Kashmir War [ceasefire ordered - Pakistan remained in a stronger position. 6th Sept celebrated as Defense of Pakistan Day]
-->1971 Fall of East Pakistan [aided by Indian Forces]
-->1998 Kargill Operation [in Kashmir region]
CORE CONFLICT ISSUES
--> Kashmir Issue
--> Siachen Issue [Siachen Glacier is the world's highest battlefield and part of the Kashmir-Tibet region]
--> Disputed dams being built by India in the Kashmir Valley that would stop water critical for irrigation in Pakistan.
--> Other trivial issues like war prisoners, maritime prisoners including fishermen, etc.