Diplomatic passports are a special form of document that is issued to foreign service officials who are engaging in work-related travel. They have special requirements and privileges that are not attached to the standard passport, or even the service passports that can be issued to government employees engaged in work-related travel. The diplomatic passport is specifically for diplomats and consular officials.
Passports were not always required for international travel. It was only during World War I that legal documents regulating entry and exit into European countries came into being, and this was as a measure to control the entry of spies and the exit of people with specialized skills. The first international passport guidelines were promulgated by a League of Nations conference in 1920, and the standardization and refinement of passports (including diplomatic passports) has continued ever since.
A diplomatic passport is a special kind of passport that is issued to foreign service officials (diplomats, consular officials, and others) and members of their families for their use on foreign postings. Therefore, diplomats who are not on a foreign assignment do not necessary possess and travel on a diplomatic passport.
The principal benefit of the diplomatic passport is that it makes one eligible for diplomatic visas. As an employee of a foreign government conducting official business, or a member of their dependent family, a diplomatic passport holder is required to get a special visa. However, since they are on official business, they will find getting a long-term visa much easier than a private citizen expatriate or immigrant would.
Holding a diplomatic passport does not automatically confer diplomatic immunity on the bearer. That must be granted separately from the diplomatic passport. Also, it is not generally true that there are separate booths for diplomatic passport holders at airport and border crossing immigration stations. These do exist, but they are not a general or required practice.
There is a special form of diplomatic passport called a Laissez-Passer, which is issued by major international organizations like the United Nations. These come in blue (general UN employee) and red (high-ranking UN official with diplomatic immunity) varieties.