Low-intensity conflict
Official definitions: 

United States: Low-intensity conflict is defined by the US Army as:

... a political-military confrontation between contending states or groups below conventional war and above the routine, peaceful competition among states. It frequently involves protracted struggles of competing principles and ideologies. Low-intensity conflict ranges from subversion to the use of the armed forces. It is waged by a combination of means, employing political, economic, informational, and military instruments. Low-intensity conflicts are often localized, generally in the Third World, but contain regional and global security implications.
The manual also says:

... successful LIC operations, consistent with US interests and laws, can advance US international goals such as the growth of freedom, democratic institutions, and free market economies. ... US policy recognizes that indirect, rather than direct, applications of US military power are the most appropriate and cost-effective ways to achieve national goals in a LIC environment. The principal US military instrument in LIC is security assistance in the form of training, equipment, services and combat support. When LIC threatens friends and allies, the aim of security assistance is to ensure that their military institutions can provide security for their citizens and government. ... The United States will also employ combat operations in exceptional circumstances when it cannot protect its national interests by other means. When a US response is called for, it must be in accordance with the principles of international and domestic law. These principles affirm the inherent right of states to use force in individual or collective self-defense against armed attack.

Relations with terrorism: Boaz Ganor notes that scholars once labeled terrorism as "low-intensity warfare." However, this terminology has become obsolete due to the intricate nature of multidimensional warfare and the mass impact of contemporary terrorist attacks, such as the September 11 attacks.

Implementation: 

Weapons: As the name suggests, in comparison with conventional operations the armed forces involved operate at a greatly reduced tempo, with fewer soldiers, a reduced range of tactical equipment and limited scope to operate in a military manner. For example, the use of air power, pivotal in modern warfare, is often relegated to transport and surveillance, or used only by the dominant side of conflict in asymmetric warfare such as a government forces against insurgents. Artillery and multiple rocket launchers are often not used when LIC occurs in populated areas. The role of the armed forces is dependent on the stage of the insurrection, whether it has progressed to armed struggle or is in an early stage of propaganda and protests. Improvised explosive devices are commonly used by insurgents, militias and sometimes government forces such as barrel bombs in low intensity conflicts. The majority of casualties in low intensity conflicts tend to be resulting from small arms and improvised explosive devices.

Intelligence: Intelligence gathering is essential to an efficient basis of LIC operation instructions. Electronic and signal gathering intelligence, ELINT and SIGINT, proves largely ineffective against low-intensity opponents. LIC generally requires more hands-on HUMINT methods of information retrieval.

Stages: In the first stages of insurrection, much of an army's work is "soft" – working in conjunction with civil authorities in psychological operations, propaganda, counter-organizing, so-called "hearts and minds." If the conflict progresses, possibly into armed clashes, the role develops with the addition of the identification and removal of the armed groups – but again, at a low level, in communities rather than throughout entire cities.

Examples: 

Burma: Myanmar (Burma) has regularly conducted limited low-intensity military campaigns against the independence movement of the Karen people in an area of southeast Burma (roughly corresponding to a Burmese administrative region called the Kayin State), which has actively pursued independence since January 1949. While allegedly limited and low-intensity in that the territories occupied in force by central government forces are returned (as they cannot be held permanently as yet) at the end of the offensives (with the stated, but sometimes unstated, purpose of weakening the opposition and independence movements), human rights organizations and national governments outside of Burma question the veracity of, and sometimes outright refute, these claims.

Sudan: The governments of Sudan have also engaged in limited military offensives (analogous to Burma's "annual dry season offensives") against various armed opposition and independence movements, which have often escalated into full-scale warfare, particularly in the south and Darfur, but also until recently in the east. These military actions (First Sudanese Civil War and Second Sudanese Civil War) have, over time, continued to ravage the areas in dispute and contribute greatly to the poor conditions in those regions as well as the various human rights violations that have occurred (and in some cases are still occurring) there.

German occupation of France: German occupation of Western Europe during World War II, notably the occupation of France, shared many aspects with more recent cases of LIC, such as the "Hearts and minds" stage early on, establishment of puppet governments, strong propaganda aimed at isolating resistance movements, and support to domestic friendly forces (such as the Milice in France).

German occupation of Poland: In Poland from 1939 to 1945 there was a strong partisan movement. Partisan forces (mainly AK and BCh organizations), although less numerous than the German army, organized a strong resistance movement; in the years 1941-44 a successful action was carried out against the expulsion of Poles from the Zamość region. Besides, the "Polish underground" destroyed hundreds of German transports of military supplies throughout the war. In Poland there was also a secret order and many non-military resistance organizations like "Zegota" which helped thousands of Jews save their lives. When the Red Army entered Poland in 1944, the Poles wanted to support them in the fight against the Germans, but the Soviets betrayed them, even though during Operation Tempest, the partisans significantly accelerated the Russian attack, the Russians arrested or killed thousands of members of the Polish Underground State, nor did the Soviets also help support the Warsaw Uprising. In total, throughout the war, hundreds of thousands of people (up to 700,000) served in the ranks of the Polish underground, and even every sixth Pole helped polish partizants but partizantes did not have more than 50,000 firearms.

Northern Ireland: The conflict, known as The Troubles, was a sectarian and ethno-nationalistic conflict, fuelled by historical events and longstanding oppression by the UK's military and security services. By the mid-1960s, the Northern Ireland civil rights movement began organizing Irish Catholics to protest, among other factors, disenfranchisement, abuses of power such as discrimination in the housing and job markets perpetuated by the ruling governments in the United Kingdom and its devolved subsidiary, known as Stormont. When these peaceful protests were met with brutal attacks by both the police and loyalist gangs given free rein to attack these protesters. On its face, it had a religious dimension although despite use of the terms 'Protestant' and 'Catholic' to refer to the two warring sides, it was not a religious conflict.  For most, these were mostly just terms of identity. A key issue was the status of Northern Ireland. Unionists and loyalists, who descended from colonists who arrived during the Ulster Plantation, displacing all but a handful of native clans and farmers, were Ulster Protestants and wanted Northern Ireland to remain within the United Kingdom. Irish nationalists and republicans, who were mostly Irish Catholics, wanted to end the Partition of Ireland, leave the United Kingdom and reunite with the 26 counties that had formed the Republic of Ireland following partition.
The main participants in the Troubles were republican paramilitaries such as the Provisional Irish Republican Army (IRA) and the Irish National Liberation Army (INLA); loyalist paramilitaries such as the Ulster Volunteer Force (UVF) and Ulster Defence Association (UDA); British state security forces such as the British Army, Royal Ulster Constabulary (RUC), Ulster Defense Regiment, MI5, and lesser known groups like the Force Research Unit; and political activists. The security forces of the Republic of Ireland played a smaller role. Republicans carried out a guerrilla campaign against British forces as well as a bombing campaign against infrastructural, commercial and political targets. Loyalists attacked occasionally republicans/nationalists, but focused primarily on the wider Catholic community in what they described as retaliation. At times, there were bouts of sectarian tit-for-tat violence, as well as feuds within and between paramilitary groups. The British security forces undertook policing and counter-insurgency, primarily against suspected republicans.  This included the internment without trial of anyone accused of being, or supporting, Republicans. Investigations also revealed significant collusion between British state forces and loyalist paramilitaries, and furthermore loyalist paramilitaries such as the Glenanne gang included serving members of the Ulster Defense Regiment and Royal Ulster Constabulary.

Guerrilla warfare: the main challenge to low-intensity warfare: Low-intensity warfare's main opponent is the guerrilla, or irregular fighter. This opponent may be state sponsored, or private non-state actors driven by religious or other ideology in urban, semi-urban and rural areas. Modern guerrilla warfare at its fullest elaboration is an integrated process, complete with sophisticated doctrine, organization, specialist skills and propaganda capabilities. Guerrillas can operate as small, scattered bands of raiders, but they can also work side by side with regular forces or combine for far-ranging mobile operations in squad, platoon or battalion sizes or even form conventional units.