The Gundam's appearances vary from series to timeline. The very first appearance occurred in "Mobile Suit Gundam 0079" which took place in the Universal Century, which has by far the largest number of appearances.
Universal Century Appearances
Mobile Suit Gundam
In the Year 0079 the "Gundam" was a name for the Earth Federation's new prototype weapon. It was a mobile suit RX-78-2 Gundam, armed with an onboard computer that collected battle data and theoretically increased the pilot's capability of operating the machine. It's main purpose was to be an answer against the Zeon mobile suit, the Zaku. Many piloted the Gundam, but the central pilot was a young civilian boy named Amuro Ray.
Gundam 08th MS Team
During the "One Year War" the Earth Federation tried to mass produce the Gundam, but this turned out to be next to impossible as it was found to be extremely expensive to produce just one; however, the symbol of the RX-78-2 Gundam, piloted by Amuro Ray of the White Base, presented too much of a morale boost to the Earth Federation troops, and they wanted to increase that by having as many Gundams on the field as possible.
In response they created RX-79[G] which was made mostly from spare parts of the original RX-78 Gundam, and in nature was only a Gundam in appearance. These Gundams were damaged often, and parts where scarce, which often ended up in using parts from other suits or completely redesigning it from the ground up. One such example of this is the RX-79[G] Ez8.
Mobile Suit Gundam 0080: War in the Pocket
Near the end of the "One Year War" the RX-78-2 Gundam was becoming increasingly inefficient. This led to the project development of what would be come known as the NT-1 Alex Gundam. As it was rumored that Amuro Ray was a "Newtype", an enlightened human being with telepathic senses, the NT-1 Alex Gundam was developed in order to suit a Newtype's capability and reflexes in battle, and was meant to replace Amuro's RX-78-2 Gundam; however, due to the actions and sacrifice of the Zeon rookie pilot, Bernard Wiseman, the NT-1 was destroyed and the entire project was scrapped.
Mobile Suit Gundam 0083: Stardust Memory
During this time the Gundam has become the Earth Federation's staple mobile suit for their designated Ace pilots. An engineering organization named Anaheim Electronics created a contract to create three Gundam prototypes for various battle situations. This resulted in the creation of the Gundams GP-01 "Zephyranthes Gundam", GP-02 "Physalis Gundam", and GP-03 "Dendrobium Stamen". Each Gundam was for a different purpose. Zephyranthes was an all-purpose machine, Physalis was built for Nuclear Warfare, and Dendrobium was built to be used for outer space battles.
The GP-02 was captured and used by the Zeons against the Earth Federation. Following the obliteration of an entire fleet of Earth Federation ships when the Zeon pilot of the GP-02 launched the nuclear payload, the GP-01 and GP-02 locked in combat and both models were destroyed. The GP-03 was then built to reaffirm the place of a Gundam at the head of an Earth Federation fleet. The GP-03 could dock with a large weapon platform codenamed "Dendrobium Orchis" and could easily hold its own against multiple large and small enemy units in the process.
Mobile Suit Zeta Gundam
After the events of Operation Stardust (Gundam 0083) the Earth Federation created a subfaction known as the TITANS whose sole purpose was hunting down the remnants of Zeon. One of the mobile suits developed RX-178 Gundam Mk II which turned out not to be much better than it's predecessor the RX-78-2 Gundam. There where three Gundam MK II's developed and on was commandeered by a young civilian newtype named Kamille Bidan. The Mk II was later upgraded to dock with a small spacecraft called the G-Defenser, which bettered the Mk II's abilities by a sizable amount.
Soon the MSZ-006 Zeta Gundam was created, its plans developed by Kamille, when he defected to the AEUG. The Zeta had an alternate mode, transforming into a jet fighter complete with wings, which made it suitable for atmospheric battles. The Zeta Gundam, along with the Mk II and the Hyaku Shiki represented the most formidable units of the AEUG, and were responsible for the downfall of the Titans and the Jupiter forces during the Gryps Conflict, and kept at bay, the Zeon army commandeered by Haman Karn.
Mobile Suit ZZ Gundam
After the events of the Gryps Conflict Kamille Bidan fell into a coma. The MSZ-006 Zeta Gundam was then taken over by another civilian newtype named Judau Ashta. As time passed, the AEUG (Anti Earth Union Group) was in desperate need for a better weapon to fight against the forces of Neo Zeon, a new enemy from the asteroid belt of the milky way made up of radical Zeon remnants. The MSZ-010 ZZ "Double Zeta" Gundam was created as an answer to this. It was much faster than the Zeta, and had more powerful weapons available in various upgrades. The ZZ also reverted to the "transform and combine" system used by the RX-78, separating into 3 individual units - core top, core base and neo core.
The restored Hyaku Shiki, Gundam MK II, Zeta and ZZ together formed the Gundam Team and served as a single unit at times.
Mobile Suit Gundam Char's Counterattack
Amuro Ray needed a new powerful mobile suit to combat against Char Azanable's powerhouse mobile suit the MSN-04 Sabazi. Amuro's personally designed RX-93 Nu Gundam was Amuro's answer. The Nu Gundam utilized Newtype weaponry such as fin funnels (psychically controlled energy beam weapons), and prototype newtype technology, the Psychoframe. The RX-93 proved to be a formidable force to fight, and easily held its own against the Sazabi and other bulky Zeon mobile suits.
However, Char felt his defeat near, and tried to steer the Axis Fortress towards the Earth, knowing that the resulting shockwave and explosion would put the Earth in a deep-freeze, allowing it to recover from the harmful effects of man's progress. The Nu Gundam's tenure was cut short as it along with it's pilot, Amuro, disappeared in a massive wave of Psychic energy that was used to prevent the Axis space fortress from falling onto the Earth.
After the events of Char’s Counterattack, a young boy named Banagher Links came into contact with a new Gundam: the RX-0 Unicorn. The Unicorn was the latest advancement in the Newtype program. Though initially appearing as a standard mobile suit with a singular horn, it had a special software called the NT-D program. When activated the NT-D caused the armor to split along various seams. This caused the mobile suit to grow in height, and a red glow was emitted from the spaces. The head also transformed, causing the single horn to split forming the traditional V-fin and the face rotated, revealing the traditional masked face of the Gundam. When in this form a newtype's abilities were enhanced allowing for greater reflexes, and enhanced telepathic like abilities that many Newtypes exhibit. The suit itself increased its output in the form of raw speed and power.
Following a battle inside one of the colonies, the RX-0 was revealed. It succeeded in impairing the NZ-666 Kshatriya of the Zeons, and was later housed inside the Nahel Argama. However, following a battle with the MSN-006 Sinanju, the Unicorn was captured. It was later freed but hurtled down onto the Earth and is currently engaged in atmospheric and ground battles.