From Dragon Ball Encyclopedia, the ''Dragon Ball'' wiki
Saiyan Armor (バトル スーツ, Batoru Sūtsu; Literally meaning "Battle Suit") is a standard-issue battle fatigue shown throughout Freeza's Force. It is made of an unknown material which is said to be stretchy yet hard. (Vegeta once remarked "one size fits all".) Even Son Gohan is able to wear Saiyan Armor that is meant for adults without having any size issues. They are known to shatter when dealt enough damage, however, showing that they are indeed hard and solid.
The most commonly seen model of Saiyan Armor had dual shoulder guards and some form of skirt armor or crotch guard. Matching boots (often white in color, but others did exist) were standard issue, as was a pair of white gloves or fingerless wrist guards. Standard Saiyan Armor had large shoulder plates, but some simply had reinforced shoulder straps. Many also had skirt armor or plate shielding the groin. Most Saiyans wrapped their tails around the armor's waist, appearing as if they had a furry belt. Some Saiyan Armor featured capes either pinned to the shoulder guards, such as that of King Vegeta and Cold, or extending from the back of the lower half of the armor, such as that of Zarbon. Saiyan Armor for women did exist, but this was only seen in Dragon Ball Z: Bardock: The Father of Goku when Selypa and some other Saiyan women in a bar wore it, but none of them wore it in the manga. Their Saiyan Armor did not include the crotch guards or the shoulder plates and looked like Vegeta's jumpsuit, but had one strap, same body armor, knee pads with a covering on one leg, and white gloves and boots.
Colored Saiyan Armor model
Vegeta wore colored Saiyan Armor when he travelled to Earth during the Saiyan Saga. When he actually arrived on Earth, the Saiyan Armor was much more like the Saiyan Armor Nappa wore in the next image. The only thing that set it apart from regular Saiyan Armor was the coloring. It should be noted that this only occurs in the anime, as at the time the scene was being animated, no official colored scenes of Vegeta or Nappa had been introduced in the manga, therefore Toei Animation had to make up their own color schemes. In Dragon Ball Kai, Vegeta's colored Saiyan Armor was changed to the normal Saiyan Armor to avoid errors with canonicity.
Bulma's Saiyan Armor model
This was a model of Saiyan Armor that Bulma Brief made for Son Goku, Gohan, Vegeta, and Future Trunks. She actually made them for all of the Z Warriors, however, Kuririn had been in a rush to find Artificial Human No. 18 when offered the Saiyan Armor and had no time to put it on. Piccolo refused to wear Saiyan Armor because he was a Namekian and "not a Saiyan", and therefore did not want to dress like one. Tenshinhan refused as well, stating that he would never wear the same outfit as Vegeta. Bulma apparently failed to offer the Saiyan Armor to Yamucha and Chaozu. They all wore a blue jumpsuit under Bulma's Saiyan Armor which covered the whole back from the neck down. The Saiyan Armor was again white with a yellow section, but only covered the chest. The shoulder guards were now nothing but two little straps made of the same yellow material. White gloves and yellow-tipped boots were also worn with Bulma's Saiyan Armor. Bulma's Saiyan Armor was mostly worn in the Hyperbolic Time Chamber and, considering the time this outfit was worn for and how much training was done in them, they held up well. After the Cell Games, Vegeta kept his blue jumpsuit (without sleeves), white gloves, and boots as his standard battle fatigue without the chest piece, possibly due to the Tenkaichi Tournament's restrictions on armor.
Saiyan Armor in a similar style was also worn by members of the Planet Elite Force despite Saiyan Armor being invented long after their deaths. Note that since Bardock and his crew were low-class warriors, they might have modified their armor in the field to suit their needs, unintentionally recreating the look of the newer models of Saiyan Armor.
Other Saiyan Armor models
During the Captain Ginyu Saga, Vegeta changed armor (as his previous one was thoroughly damaged). The armor he replaced it with featured no shoulder guards and instead had straps that simply extended the white part of the armor. Vegeta stated that this was an older model.
Paragus wore a unique model of Saiyan Armor in Dragon Ball Z: Broly: The Legendary Super Saiyan. The overall design resembled the clothing of those on Planet Yardrat. His gloves and boots were purple throughout most of the film, though in a few flashbacks his gloves, boots, and armor were differently colored, i.e. yellow/orange.
In Dragon Ball Z: Cooler's Revenge, a model of Saiyan Armor was the customary battle fatigue of soldiers in Cooler's Armored Squadron. Extended length of brown armor jutted from their left shoulder, while the uniforms themselves were in Cooler's primary colors of purple and white, although the entire armor was lime-yellow. The Saiyan Armor was somewhat of a hybrid between the older model Vegeta wore during the Captain Ginyu Saga and the standard model (featuring a shoulder guard and a plain strap). Interestingly, the insignia resembled Cooler's head in his Final Form.
It is named Saiyan Battle Armor in Dragon Ball Z: The Legacy of Goku II.