Weapons and armour historically interact in rather consistent ways. But fantasy often completely ignores this. It should be noted however that it is usually armour development – itself a result of development in metallurgy – that forces emphasis on a certain type of weapons; weapon development rarely drives armour development up until the appearance of gunpowder weapons.
No armour or shield
In this case, anything goes. But primary weapons will be ones which allow dealing enough damage 1) quickly, and 2) at sufficient distance. This means bows, slings, javelins and spears.
No armour + shield
Shield is most useful in group combat, such as phalanx. As such, spears will be extremely significant – either long thrusting spears or else javelins. Sword will be a backup weapon, and usually relatively short so as to allow thrusting and maneuvering during shield push. Archers are a significant danger, though their effectiveness can be significantly reduced with proper tactics.
Weapons themselves will have relatively broad blade in order to allow effective cutting and thrusting both – broadsword, daggers with broad blades, combat knives etc.
Textile armour + shield
Textile armour (gambeson) is easily pierced by arrows and defeated by blunt weapons. As such, person using solely textile armour will want to use a shield, and one capable of protecting them from missile fire. However, cutting and piercing weapons have to defeat several layers of very thick and resistant cloth to reach body, and air pockets created in the quilting help absorb the shock of impact. As such, swords used along with textile armour will already be predominantly piercing-oriented, though cutting blades such as falchion are not unimaginable.
If worn under plate armour it is called arming doublet (aketon).
Mail armour + shield
Mail armour of significant extent requires weapons to be optimized for piercing (in order to split the rings), as mail renders cutting useless. Swords and daggers will thus become more pyramidal in shape, and primary weapons will be spears, axes, maces, morningstars and similar. Spears themselves will change in design, with narrower and more pointed spearheads to defeat armour. Couched lance may be necessary to ensure penetration, but most of the time even a well-placed powerful thrust from a spear or lance (e.g. using overarm technique) will still defeat the armour.
Arrows are still dangerous, especially if armour does not provide full coverage. Bows and crossbows have to be much stronger than normal hunting weapons in order to defeat mail, but are fully capable of doing so, thus requiring shield to be worn. This does depend on metallurgy and overall defences: padded jack over mail haubergeon may render the wearer immune to Mary Rose longbows; but such high-quality mail is likely to only be available at the point when metallurgy has allowed the development and widespread utilization of plate armour.
Partial plate has solid plates with mail covering the gaps between the plates. As such, swords and daggers will be optimized for piercing mail, as will spear heads. Blunt weapons such as warhammers and morningstars increase in importance, though plates leave significant enough areas uncovered for stabbing swords and daggers to still present a significant danger. Both of these will be shaped geometrically, like a very slim pyramid, in order to get a slim piercing point that can slip between the plates and through mail. Half-swording and murder strike become standard swordfighting techniques. Polearms will likewise get pointy and stabby, with broad leaf-shaped points replaced with slim geometrically-shaped spikes capable of slipping between the plates and through the mail. Arrows are still somewhat of a danger as they can hit between the plates, though combination of mail and arming doublet (aketon) can still stop them.
Shields become smaller, but still extant, as plate coverage still leaves some crucial areas protected only by mail.
Plate armour makes wearer all-but-immune to bladed weapons. The only danger are very thin piercing points which can slip between plates and through mail, blunt weapons, and combination of the two (as in warhammer beak).
As a result, when plate armour becomes widespread, primary weapons will be warhammers and polehammers (and pollaxes), with either hammers for dealing blunt damage or spikes for punching through plate. Secondary weapons will be arming sword and dagger, with blades that will achieve fully geometrical forms if that has not happened already. Polearm points get even more pointy.
Shields mostly disappear. Pavises may be used to protect more-lightly-armoured troops and prevent lucky hits.
Angle of attack matters much more with full plate armour than with other types, as solid plate does not provide opportunities for weapons to “bite” into armour. Between strength of plate and curvature, arrows are reduced mostly to annoyance and eventual lucky hits. Even a 1 000 lbs crossbow will not pierce mediocre plate. What can happen is that bolt or arrow may warp and thus defeat armour at joints, vision slits and breathing holes.