Factions and equipment
The military forces of the Dragon World are seen using forces very similar to that of the real world's militaries. Infantry men are of course the main faction, and use such equipment as different types of guns, grenades and other explosives, bazookas, and much more. In terms of vehicles; tanks, jeeps, and armored cars seem to be used. The military is also composed of some sort of Navy or Air Force, as large battle ships and high powered jets are also employed. The King's Guard is a main faction within the military, most notably seen during the King Piccolo Saga and later during the Cell Games Saga. There are different types of Town Militia, which are factions within the main military force as well.
It is unknown exactly why such a force exists in the Dragon Ball world, as Earth at the time of the main series is fully governed by one king, so therefore it is unlikely that the same one nation would go to war with itself (although the fact that the Earth has a station broadcasting international news implies that the Earth may have more than one nation). However, it is possible that this military force, other than just for simple precaution, has existed long before Earth's government had become one. Another theory is that it was developed during King Piccolo's reign of destruction, and has been kept ever since in case of another emergency such as that. It is plausible that this military is mainly used to maintain peace, by combating such organizations as the Red Ribbon Army.
However, during Dragon Ball Z, Earth's Military forces are called into action on numerous occasions. When the two Saiyans Vegeta and Nappa threaten Earth, the Northern army does prepare to attack, only to be completely decimated by Nappa during his rampage.
When Cell kills and absorbs all of the civilians in Gingertown, the military arrives with an armada of tanks, only for them to be destroyed by powerful energy beams from the monster's eyes. Later, when Cell threatens to destroy the Earth if it is not defeated in his tournament, King Furry sends his Royal Guard in an attempt to destroy him again, this time believing he had brought more than enough infantry and vehicles, only to be defeated with ease once again.
Seven years after Gohan defeats Cell, the military is once again drawn into action with the threat of Majin Buu looming over the world. Despite an overwhelming force of battle ships and jets, they are no match for the power of Majin Buu and are effortlessly destroyed yet again.
Video game appearances
The King's Guards appear in the Cell Games Arena battle stage in the arcade game Dragon Ball Z 2: Super Battle, and they are seen in the city Goku visits right before the battle against Nappa and Vegeta in Dragon Ball Z: The Legacy of Goku.