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troperville

It started from the Guomindang forces trained by Chiang Kai Shek during the 1920s, during which they initiated the Northern Expedition of 1926-28. Later, they participated in the war to wipe out the Communist guerrillas of Mao Zedong, but it was later proved to be punitive. When Imperial Japan later attacked China during the Manchurian Incident in 1931, the National Revolutionary Army, then known as the NRA, were screwed. It is also the same case in the Second Sino-Japanese War, though fairly, they are never willing to give up. Unfortunately, the Nationalists squabbled a lot with the communists, and as the Communists later curbstomped the Nationalists in 1949, most of the ROC later relocated to Taiwan.

The ROC Armed Forces was later improved with US weaponry. Previously, the ROC had captured Japanese weapons, and many of their guns were purchased from Nazi Germany [though eventually, the ROC used US weapons almost exclusively].

The ROC allegedly sent some special forces against the Chinese Communists in the Korean War through Burma, but this was never proven. They also sent advisors to Vietnam.

The Taiwanese also sent mercenary pilots with their F-5s into North Yemen to fight South Yemen. Incidentally, PRC helped North Yemen too.

The ROC military is composed of the Army, the Navy, the Marines, the Air Force, and of all cases, the Military Police. The Coast Guard is not under the jurisdiction of the Ministry of Defense but may be attached in wartime.

Conscription was the rule; every 18 year-old, able-bodied male had to serve. In the last few years, the term of service has been regularly shrinking, however...

Tropes embodied:

Badass Army: Current projections by both the US and Chinese governments that were leaked by wikileaks reveals that they both think that the tiny little island nation's military could easily repel a massive Chinese invasion while inflicting heavy losses due to superior technology, morale, defenses, and China's overall lack of a navy.
Butt Monkey: the Nationalist Army almost always loses out in the early stages of the Sino-Japanese War.
- Although part of it is deliberate; both Nationalists and Communists have been independently agreed on the fact that the China would have lost to Japan on a face-to-face combat; their strategy is to let Japanese take as much land in the beginning such that the Japanese military would be stretched too thin, and the Chinese would have time to prepare a more realistic rebound. As a result, those battles were intended to delay not deter; and before the entrance of the US they used a combination of terrorist and guerrilla tactics to ensure the Japanese would be stretched thin.
Conscription: Like all draft-based armies, this is the cause of its share of problems and animosity. It goes back to the transformation from the National Revolutionary Army to Republic of China Armed Forces—officers were almost exclusively (professional) Chinese evacuees from the Civil War, while enlisted men were overwhelmingly (poor) Taiwanese conscripts (an racial divide that unfortunately survives to this day, though is far less severe).
- Additionally, across the strait the other other Chinese Army gets its manpower from volunteers (making official conscription redundant), as oppose to requiring men go before a formal draft board. Unlike it's first-rate technology, the branches of the ROC Armed Forces probably cannot count on first-rate morale: hence the shrinking length of mandated service.
Putting on the Reich: A suprisingly Allied example, with their Stahlhelm helmets. Justified in that Germany trained the Nationalist Army in the 1930s. It doesn't help that some of the ideology of the Kuomintang had fascist overtones and Chiang for a time expressed his admiration for fascism. The Japanese invasion changed his view, of course.
Took a Level in Badass: Ironically the ROC forces started to become competent after they were exiled to Taiwan.
What the Hell, Hero?: When the ROC forces arrived in Taiwan they forcefully took over every factory they could find, including the sugar factories (which had been built by Imperial Japan). Many Taiwanese were beaten up, jailed, or outright slaughtered by the Nationalists for protesting against the ROC Army's conduct. If the Taiwanese were poor under the Japanese, they were now destitute and starving under the rule of the Nationalists. This terrible scenario changed when America installed naval bases later and (presumably) demanded changes in attitude. One wonders if Chiang Kai Shek was terrible at managing his soldiers' behavior towards civilians...

In fiction:

In Zhang Yimou's To Live, the main character joins the Nationalist Army during the Civil War. When he and his friend sleep through a battle in which the Communists wipe out their entire unit, they surrender to the Communists, who let them join up and live in exchange for providing entertainment with their traditional Chinese shadow puppets.
Chu Yen-ping's Sige Buping Fande Shao Nian (Forever Friends) is a heartwarming 1996 comedy that deals with the less-than-professional behavior not unheard of in recent volunteers and conscripts in the ROC Army.
In the Web Original Alternate History Superpower Empire China 1912 , they appear to be more competent. In fact, China at this point went back to the Imperial system, albeit modernized. They still use the Republic of China flag since the founding Emperor at the new dynasty is a Han scholar who despite their ideological differences, is a friend of Sun Yat Sen.