Battle Armor (called Saiyan Armor by the Earthlings before the Namek Saga) is standard-issue combat attire throughout the Planet Trade Organization. They are made of an unknown material, which is said to be stretchy, yet durable (Vegeta once remarked "one size fits all"). Even Vegeta as a Great Ape was able to wear his armor without having any size issues. They are known to shatter when dealt enough damage, showing that they are indeed hard and solid.
The most commonly seen ones have dual shoulder guards and some form of skirt armor or crotch guard. Matching boots (often white in color, but others do exist) are standard issue, as is a pair of white gloves or fingerless wrist guards. Standard armor has large shoulder plates, but some simply have reinforced shoulder straps. Many also have skirt armor or plate shielding the groin. Most Saiyans wrap their tails around the armor's waist, appearing as if they have a furry belt. Some armors feature capes, either pinned to the shoulder guards such as that King Vegeta and King Cold, or extending from the back of the lower half of the armor, such as Zarbon has. Battle Armor for women do exist too, but this is only seen in the Bardock special as Fasha and some women in a bar are wearing it, and none are seen wearing it in the manga. It does not include the crotch guards or the shoulder plates, and looks like Vegeta's jumpsuit, but having one strap, same body armor, knee pads with a covering on one leg, and white gloves and boots.
We see Vegeta wear this when he is traveling to Earth in the Saiyan Saga. When he actually arrives on Earth, the armor is much more like the armor Nappa is wearing in the next image. The only thing that sets it apart from regular Battle Armor is its coloring. It should be noted, though, that this only occurs in the anime as, at the time the scene was animated, no official colored scenes of Vegeta or Nappa had been introduced in the manga, so Toei Animation had to make up their own color schemes. In Dragon Ball Kai, Vegeta's colored armor is changed to the normal armor, to avoid errors with canonicity.
Bulma's armor model
This is a variation on the above armor that Bulma makes for Goku, Gohan, Vegeta and Future Trunks. She actually made suits for all of the Z Fighters; however, Krillin was in a rush to find Android 18 when offered and had no time to put it on. Piccolo refused to wear one because he was a Namekian, "not a Saiyan" and did not want to dress like one. Tien Shinhan refused as well, stating that he would never wear the same outfit as Vegeta. Bulma apparently failed to offer the armor to Yamcha and Chiaotzu. They all wear a blue jumpsuit under the armor that covers the whole back from the neck down. The armor is again white with a yellow section, but only covers the chest. The shoulder guards are now nothing but two little straps made of the same yellow material. White gloves and yellow tipped boots are also worn with the outfit. We mostly see this armor 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 keeps his blue jumpsuit (without sleeves), white gloves, and boots as his standard battle outfit but loses the chest piece, possibly due to the World Tournament's restrictions on armor.
It is in the Cell Games Saga where the defensive properties of the Battle Armor are best displayed. Prior to this saga, Battle Armor is often seen destroyed, seemingly easily, by the simplest blasts (even a Special Beam Canon with power level less than 1,500 was enough to damage Raditz' armor), casting doubts as to whether the armor was genuinely effective at protecting its wearer. However, in the Cell Games Saga, by Vegeta's own admission, in the episode "The Fight is Over", he was weaker than Goku. Despite this, Goku was beaten to a pulp by the Cell Juniors, and was unable to even defend himself, whereas Vegeta and Trunks (the only two wearing Saiyan Armor) were the only ones able to stay on their feet, despite the thrashing that they took from the Cell Juniors. This side-by-side comparison effectively demonstrates that, although the armor is often seen being destroyed in battle, when it remains intact, it still provides a good deal of protection.
Armor in a similar style is also worn by members of Bardock's Elite, despite its 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.
During the Captain Ginyu Saga, Vegeta adopts new armor to replace his previous one due to it being thoroughly damaged. The new armor features no shoulder guards and instead has straps that simply extend the white part of the armor. Vegeta states that this is an older model.
In Dragon Ball Z: Cooler's Revenge, Cooler's Armored Squadron sport armor that combines traits of both the older armor model worn by Vegeta during the Captain Ginyu Saga and the new, standard armor model. Notable differences with this "hybrid" armor include a lime green coloration instead of white or black and the Squadron's insignia (which interestingly resembles Cooler's head in his final form) fully covering the abdominal portion of the armor.