This seems to be one of those subjects that are always controversial in the realm of weapons, warfare, and armor. First of all we need to ask ourselves what circumstances we are looking at:
1. One on One combat (judicial duel, etc...)
2. Campaign Warfare (cavalry, mounted warfare, infantry, troop movements, formations, breeches and so forth)
Now what kind of weapons were common in these circumstances:
1. Polearms: spears, halberds, pollhammers, pollaxes
2. Impact weapons: Axes, maces, war hammers
3. Swords: single handers, longswords, 2-handers (Great Swords, War Swords)
Armor:
1. Plate Harness
2.
Chain mail
3. Lamellar
4. Hybrid of previous types (breast plate, arming jacket, lamellar and chain mail mixes)
Depending on the period all of the above could apply. Cutting weapons were not very useful against heavier armor, even chain mail was a pretty good defense against all but the most committed cuts. Thrusting actions were good against chain mail, especially with a reinforced or very tapered tip, and spears, thrusting swords, picks, lances, spikes on pollaxes, etc... were developed for this function. Plate mail is even harder to get through, you either need a huge amount of force to literally puncture it (pollaxe spikes and the like were probably capable of actually puncturing the plates themselves, but this is a matter of debate and the guy isn't usually going to stand there and let you spike him) or you need something capable of getting between the plates or in the small crevices and uncovered areas like the armpit, groin, behind the knee, underneath the wrist, and so on, and even these were usually covered by chain mail so we are back to the previous step. If you have a heavy impact weapon, a halberd, pollaxe, pollhammer, or mace, you can then disable the armored opponent by either screwing up the articulated joints of the armor or just breaking bones, causing contusions, or smashing the skull apart through the helmet. Even with these fearsome weapons there was a lot of grappling, joint locks, and throws going on coupled with a heavy dagger (usually a rondel) to poke through the gaps in plate and through the underlying chain mail. A knight or man at arms in full harness was not invincible, but he was impervious to a lot. Not everyone could afford or even wanted to wear all that armor, however, and there were plenty of people running around much more lightly armed and armored. Of course, we are leaving out arrows and crossbow bolts out for now, which were a huge part of warfare during this period.
Warfare from the Dark Ages to the Renaissance was a mutable and constantly changing entity, and there was a serious arms and armor race on as technology and craftsmanship was pushed to the limits to counter one another. Most of the manuscripts and treatises regarding combat during these periods made distinction between combat in warfare and in the street or for duels, and much of that was based around the vulnerability of the opponent (what armor he was or wasn't wearing) as well as the circumstances surrounding the combat. What worked for a knight in the 13th Century was not really well suited to the 15th Century, and so on.