Valyrian steel, although being often quoted as being 'far too rare', nevertheless seem to be found pretty much at every armoury in Westeros... ;-)
Besides that, I always imagined that Valyrian steel shares the characteristics of Japanese steel in the Middle Ages which is tempered steel made of many layers, by repeatedly folding and hammering on the steel that will produce the blade part (see https://en.wikipedia.org/wiki/Japanese_swordsmithing). Also, Japanese traditionally used several different kinds of steel in producing the famous Samurai swords; the actual workmanship of such a sword (even today) is incredible, for the sheer number of processes involved in doing the whole sword (and not just the blade itself).
This would be consistent with a technique that is primarily used for swords and/or edged weapons, since besides lightness (to make them easier to swing), you need to avoid brittleness, you need hardness for the edge, it must be flexible to an extent but not too much, you should be able to polish it and etch it with designs (for aesthetic reasons), without, however, endangering the lightness (polish/etch it too much, and you ruin the sword), and so forth — an impressive amount of different characteristics that are usually mutually exclusive. The Japanese dealt with those by employing several different steel types. I'm no metallurgist, but the actual Japanese blade is more like a 'steel alloy' — a combination of different kinds of steel welded together — than a single kind of steel.
Such techniques in Japan were sacred (involving many Shinto rites) and passed very ceremoniously from master to apprentice (and apprenticeships would be very, very long), and, as such, there would be an almost mystical aura involved in those swords. Even today, the ancient art of forging traditional swords has not been lost, and there are still masters doing them, exactly like they have been made in the past; such masters are still revered for their forging skills, and the transmission of knowledge to an apprentice is still enshrouded in mystery, secrecy, and religious rites.
Japan, in the Middle Ages, did not trade directly much with Europe (and most certainly they wouldn't be selling their Samurai swords!), or the Europeans would think of Japanese swords (assuming they could use them for fighting at all, since they also require special techniques to wield, unlike European swords — basically, point the sharper end towards the enemy and hit him with it ;) ) with the same kind of reverence that the inhabitants of Westeros talk about Valyrian steel.
Such techniques for forging swords would probably be useless for doing armour. Also — and we're sticking with real history here, of course any author is free to invent their own metallurgy rules for their universe! — different masters would create armour, and sword makers would most certainly not spoil their delicate fingers in doing armour. Therefore, it would be extremely unlikely that exactly the same kind of steel would be employed simultaneously in swords and armour. Also remember that such steel would have completely different functions — the things that they would share would just be the need to be very light. Swords are made for cutting, while armour is made for resisting to be cut — this requires completely different kinds of steel, and even if an analogous of the layered technique used by the Japanese could be employed in armour as well, it would be quite differently done (and with different kinds of steel), even though in terms of final workmanship — polishing, etching, and so forth — which would be done by a different master, ceremonial armour would very likely also be as decorated as the swords (just like in Europe in the 1500s).
An alternative to the Japanese method of making swords would be the Damascus art of making swords. Damascus-made swords are a mix of two characteristics: a special technique to forge the swords (which sadly has been lost to us) as well as a special kind of steel, that was imported from India into Damascus where it was successfully forged into especially tough swords, resistant to breaking, and capable of being sharpened to a fine edge. Even though the art of forging Damascus steel swords was lost to us by the mid-18th century, it seems very plausible to admit that such swords required both a special craftsmanship and skill as well as a particular kind of ore to produce the special steel (probably with traces of vanadium but also mixed with certain plants to give the Damascus steel blades its characteristic 'organic' patterns on the surface). If you lose either the craftsmanship or the origin of the ore, you are unable to reproduce the manufacture of such swords. This would also be consistent with the idea that nobody can forge Valyrian steel blades any more, since both the ore and the forging skills were lost during the Doom, and you require both to successfully forge swords. Damascus steel, like Japanese steel, was not appropriate for forging armour, though. It was relatively well-known in Europe — especially during the many wars and Crusades to the Middle-East — although I would think that most Middle-Age noblement would refuse to use 'infidel'-produced swords for their personal use, although later on, towards the beginning of the 18th century, I can well believe that its import might have boomed...
The alternative is to assume that what is important is the kind of steel, not the technique itself, i.e. assume that there is a special source of ore with 'almost magical' qualities, that, if correctly extracted and treated, will produce the qualities attributed to Valyrian steel. In that case, what would make Valyrian steel special would be the mines where such ore was extracted, and, of course, after the Doom, such mines would be gone, with no way to replicate Valyrian steel ever again.
This would not be a far-fetched idea, either. In ancient times, the Toledo region featured 'special' iron ores, which were used for swords with unique qualities that could not be reproduced elsewhere, no matter how good the sword maker was. Toledo steel was historically known to be 'unusually hard', even by the Romans; and the regions surrounding the iron mines in the Iberian Peninsula were already known by them for their manufacture. Toledo steel swords were widely spread all over Europe, but the actual aesthetic value of such blades would not be the crucial argument for its use, but rather the physical characteristics, i. e. how well it performed in real battles.
This is closer conceptually to what 'Valyrian steel' is supposed to look like or to be, but, of course, in this case, there wouldn't be any 'Valyrian steel armour', since Damascus-style steel would not be appropriate for doing armour... while technically Toledo-style steel might be harder (good for armour) but most certainly weighing much more than 'common' steel.
Oh, and then again, we are in the realm of fantasy anyway, so why can't we have a super steel alloy that is simultaneously good for weapons and armour? After all, Tolkien got away with mithril, and he was much more meticulous about the tiniest details in his world, why shouldn't Martin get away with Valyrian steel, especially because most of the world is incredibly incoherent anyway? :-D
