According to wikipedia, Diplomatic immunity is a form of legal immunity and a policy held between governments, which ensures that diplomats are given safe passage and are considered not susceptible to lawsuit or prosecution under the host country's laws (although they can be expelled). It was agreed as international law in the Vienna Convention on Diplomatic Relations (1961), though the concept and custom have a much longer history. Many principles of diplomatic immunity are now considered to be customary law. Diplomatic immunity as an institution developed to allow for the maintenance of government relations, including during periods of difficulties and even armed conflict. When receiving diplomats—formally, representatives of the sovereign (head of state)—the receiving head of state grants certain privileges and immunities to ensure that they may effectively carry out their duties, on the understanding that these will be provided on a reciprocal basis.
Originally, these privileges and immunities were granted on a bilateral, ad hoc basis, which led to misunderstandings and conflict, pressure on weaker states, and an inability for other states to judge which party was at fault. Various international agreements known as the Vienna Conventions codified the rules and agreements, providing standards and privileges to all states.
It is possible for the official's home country to waive immunity; this tends to only happen when the individual has committed a serious crime, unconnected with their diplomatic role (as opposed to, say, allegations of spying), or has witnessed such a crime. Alternatively, the home country may prosecute the individual. Many countries refuse to waive immunity as a matter of course; individuals have no authority to waive their own immunity (except perhaps in cases of defection).