Irregular military
Regular vs. irregular: The words "regular" and "irregular" have been used to describe combat forces for hundreds of years, usually with little ambiguity. The requirements of a government's chain of command cause the regular army to be very well defined, and anybody fighting outside it, other than official paramilitary forces, are irregular. In case the legitimacy of the army or its opponents is questioned, some legal definitions have been created.
In international humanitarian law, the term "irregular forces" refers to a category of combatants that consists of individuals forming part of the armed forces of a party to an armed conflict, international or domestic, but not belonging to that party's regular forces and operating inside or outside of their own territory, even if the territory is under occupation.
The Third Geneva Convention of 1949 uses "regular armed forces" as a critical distinction. The International Committee of the Red Cross (ICRC) is a non-governmental organization primarily responsible for and most closely associated with the drafting and successful completion of the Third Geneva Convention Relative to the Treatment of Prisoners of War ("GPW"). The ICRC provided commentary saying that "regular armed forces" satisfy four Hague Conventions (1899 and 1907) (Hague IV) conditions. In other words, "regular forces" must satisfy the following criteria:

being commanded by a person responsible for his subordinates to a party of conflict
having a fixed distinctive emblem recognizable at a distance
carrying arms openly
conducting operations in accordance with the laws and customs of war
By extension, combat forces that do not satisfy these criteria are termed "irregular forces".

Types: The term "irregular military" describes the "how" and "what", but it is more common to focus on the "why" as just about all irregular units were created to provide a tactical advantage to an existing military, whether it was privateer forces harassing shipping lanes against assorted New World colonies on behalf of their European contractors, or Auxiliaries, levies, civilian and other standing irregular troops that are used as more expendable supplements to assist costly trained soldiers. Bypassing the legitimate military and taking up arms is an extreme measure. The motivation for doing so is often used as the basis of the primary label for any irregular military. Different terms come into and out of fashion, based on political and emotional associations that develop. Here is a list of such terms, which is organized more or less from oldest to latest:

Auxiliaries – foreign or allied troops supplementing the regular army, organized from provincial or tribal regions.  In the Imperial Roman army, it became common to maintain a number of auxiliaries about equal to the legionaries.
Levies – feudal peasants and freemen liable to be called up for short-term military duty.
Privateer – a "for-profit" private person or ship authorized and sponsored by a government by letters of marque to attack foreign vessels during wartime and to destroy or disrupt logistics of the enemy during "peacetime", often on the open sea by attacking its merchant shipping, rather than engaging its combatants or enforcing a blockade against them.
Revolutionary – someone part of a revolution, whether military or not.
Guerrilla – someone who uses unconventional military tactics. The term tends to refer to groups engaged in open conflict, rather than underground resistance. It was coined during the Peninsula War in Spain against France.
Montoneras – they were a type of irregular forces that were formed in the 19th century in Latin America.
Franc-tireur – French irregular forces during the Franco-Prussian War. The term is also used in international legal cases as a synonym for unprivileged combatant (for example the Hostages Trial [1947–1948]).
Militia – military force composed of ordinary citizens.
Ordenanças – The Portuguese territorial militia system from the 16th century to the 19th century. From the 17th century, it became the third line of the Army, serving both as local defense force and as the mobilization system that provided conscripts for the first (Regular) and second (Militia) lines of the Army.
Partisan – In the 20th century, someone part of a resistance movement. In the 18th and 19th century, a local conventional military force using irregular tactics. Often used to refer to resistance movements against the Axis Powers during the Second World War.
Freedom fighter – A type of irregular military in which the main cause, in their or their supporters' view, is freedom for themselves or others.
Paramilitary – An organization whose structure, tactics, training, subculture, and (often) function are similar to those of a professional military, but which is not part of a country's official or legitimate armed forces.
Terrorist – An irregular military that targets civilians and other non-combatants to gain political leverage. The term is almost always used pejoratively. Although reasonably well defined, its application is frequently controversial.
False flag or pseudo-operations – Troops of one side dressing like troops of another side to eliminate or discredit the latter and its support, such as members of the Panzer Brigade 150, commanded by Waffen-SS commando Otto Skorzeny in Operation Greif during the Battle of the Bulge in World War II and Selous Scouts of the Rhodesian Bush War.
Insurgent – An alternate term for a member of an irregular military that tends to refer to members of underground groups such as the Iraqi Insurgency, rather than larger rebel organizations like the Revolutionary Armed Forces of Colombia.
Fifth column - A group that carries out sabotage, disinformation, espionage, and/or terrorism within a group that responds to external enemies
Bandit - It is generally treated as an organized crime, but it has the character of a resistance movement depending on the political and social situation.
Private army - Combatants who owe their allegiance to a private person, group, or organization.
Mercenary or "soldier of fortune" –  Someone who is generally not a national in a standing army or not otherwise an inherently-invested party to an armed conflict who becomes involved in an armed conflict for monetary motives or for private gain. Mercenaries are often explicitly hired to fight or provide manpower or expertise in exchange for money; material wealth or, less commonly, political power. Mercenaries are often experienced combatants or former regular soldiers who decided to sell their combat experience, skill or manpower to interested parties or to the highest bidder in an armed conflict. Famous historic examples of "professional" or organized (often "career") mercenaries include the Italian condottieri, or "contractors", leaders of "free agent" mercenary armies that provided their armies to the various Italian city-states and the Papal states during the Late Middle Ages and Renaissance Italy in exchange for profit, land or power. However, not all soldiers deemed to be "mercenaries" are "professional" or "career" mercenaries, and many mercenaries may be simply opportunists or persons with no prior combat experience. Whether a combatant is truly a "mercenary" may be a matter of controversy or degree, as financial and national interests often overlap, and most standing regular armies also provide their soldiers with some form of payment. Furthermore, as reflected in the Geneva Convention, mercenaries are generally provided less protection under the rules of war than non-mercenaries, and many countries have criminalized "mercenary activity".
Intense debates can build up over which term is to be used to refer to a specific group. Using one term over another can strongly imply strong support or opposition for the cause.
It is possible for a military to cross the line between regular and irregular.  Isolated regular army units that are forced to operate without regular support for long periods of time can degrade into irregulars. As an irregular military becomes more successful, it may transition away from irregular, even to the point of becoming the new regular army if it wins.

Regular military units that use irregular military tactics: Most conventional military officers and militaries are wary of using irregular military forces and see them as unreliable, of doubtful military usefulness, and prone to committing atrocities leading to retaliation in kind. Usually, such forces are raised outside the regular military like the British SOE during World War II and, more recently, the CIA's Special Activities Center. However at times, such as out of desperation, conventional militaries will resort to guerilla tactics, usually to buy breathing space and time for themselves by tying up enemy forces to threaten their line of communications and rear areas, such as the 43rd Battalion Virginia Cavalry and the Chindits.
Although they are part of a regular army, United States Special Forces are trained in missions such as implementing irregular military tactics. However, outside the United States, the term special forces does not generally imply a force that is trained to fight as guerillas and insurgents. Originally, the United States Special Forces were created to serve as a cadre around which stay-behind resistance forces could be built in the event of a communist victory in Europe or elsewhere. The United States Special Forces and the CIA's Special Activities Center can trace their lineage to the OSS operators of World War II, which were tasked with inspiring, training, arming and leading resistance movements in German-occupied Europe and Japanese occupied Asia.
In Finland, well-trained light infantry Sissi troops use irregular tactics such as reconnaissance, sabotage and guerrilla warfare behind enemy lines.
The founder of the People's Republic of China, Mao Zedong actively advocated for the use of irregular military tactics by regular military units.