For both armor and for weapons, metals are a fantastic choice. They offer hardness and ductility that are hard to obtain in other materials. There are, however, a few other choices worth looking at.
Leather armor, sometimes reinforced by boiling it in water or oil, makes an effective armor. It's less effective than steel, but also cheaper.
Wood is stiff and clunky, but was used by the ancient Siberians to build armor out of. Specifically, wooden pauldrons were used in addition to suits of lamellar armor made from boiled seal skin.
Armor can also be made from sheets of paper laquered together. This was used by the ancient Chinese as a form of armor, and was quite effective. It's cheaper to make than steel armor, but breaks down over time. It's had some minor implementation in the modern day, as well.
Ceramic is harder than steel. However, it's also quite brittle. It makes for some extremely sharp knives and tools, but swords, for the most part, are made to survive hard knocks. There is, however, an exception to that general rule that is quite common in fantasy settings: the katana. The Japanese didn't develop smithing to the same degree as the Europeans, so they relied on the process of folding somewhat weaker steel in order to produce a sharp blade. This produced a blade that was stiff and excellent for cutting, but would shatter if used incorrectly. Because of this, a level of mastery was needed with a katana before it could be used for war.
Ceramic offers some similar properties to the steel used in katanas. It's incredibly hard, but also brittle. Your elves could use blades with ceramic edges in a similar way that the katana was used. The focus would be on a single, powerful killing blow, with little to no sword to sword contact or parrying, which would chip or shatter the blade. Properly struck, such a blow would allow a ceramic blade to slice through metal armor more easily than a steel sword.
Long ceramic blades would be tough to make, and even tougher to make strong, so it's likely that either something more like an ax would be used, or else that a sword blade would be made from multiple, overlapping ceramic blades.
Wood, bone, and carbon fiber
Wood, bone, and carbon fiber could be used to make weapons, but don't hold an edge well. They do, however, have the advantage of resiliency over something like ceramics. They bend without breaking, making them better for something like a handle or the shaft of a spear. For a sword, carbon fiber could be used for everything but the edge, which could be made of a stronger material like a ceramic.
Other natural materials
On Earth, other natural materials with the hardness and strength to be excellent weapons, easily on par with steel, do exist. Unfortunately, they don't come in large quantities, generally found in tiny structures like the teeth of limpets, which are made from a composite stronger than anything we've ever made in a lab.
The claws or teeth of dragons, in your world, could be made of such a material. If they were, the elves could use this as their main material for weapons, making elven blades rare, but the sharpest and strongest in the world.