The "Arming Sword" is what I think most people think of (incorrectly) as a "longsword" or a "broadsword". The arming sword is the classic "knightly" sword of the High Middle Ages - a one-handed, cruciform, double-edged cut-and-thrust weapon with wide quillons and a heavy pommel.
Typically would be about 3' long, and about 2-2.5 lbs. Balance point is a few inches along the blade from the guard.
Some of these have been dispelled before, but it bears repeating. One of the main myths around swords is that they were heavy - this is not so. An arming sword would weigh a little over 2lbs - no more than 2.5lbs.
They were not the dull, heavy, metal clubs some people seem to think they were - the arming sword is nimble and sharp. Now, the sword wouldn't be sharp like a razor blade is sharp - more like a chisel, but still sharp enough to easily cut an arm or leg off, decapitate someone - even cut them right in half. I'll have some caveats to this in my longsword article.
This is more of a general combat misconception, but sword combat doesn't take place at punching distance - two people aren't going to stand 4 feet apart bashing away at each other. Much of the fight will be at more like 10 feet, with a lot of fluid forward and back movement as the fight progresses.
A misconception I saw in a comment on another one of my posts was that you could wait for a sword swing to "go by", then rush in. This is not possible, as sword swings do not "go by". The finishing position of a sword strike that doesn't connect is with the point out in front of you, directed at your opponents neck or head.
Even if the sword did "go by", the nimbleness of a sword means that your rush would simply end with you having an arm cut off or being impaled.
The European cruciform sword is one of the most versatile weapons out there. It has cutting edges on both sides, and is straight and pointed for powerful thrusts. The quillons and pommel can be used to punch and strike - some swords even have sharpened quillons for more powerful quillon punches. In a pinch, it can even be used two-handed with one hand on the pommel, although it won't have the power of a true two-handed weapon, due to the limited leverage the off-hand will have.
The arming sword can be used with a shield or buckler (as detailed in MS I.33, the earliest known European martial arts manual) or alone (perhaps more common in a self-defense situation, described in Fiore's works).
I.33 does a good job of illustrating the way you fight with sword and buckler - keeping the two together, moving them as one weapon, using the buckler to cover your sword arm as you attack. You still are mainly using your sword to defend yourself - the buckler is there to help. I'll write more about shield combat when I get to shields. Suffice to say at this point that 1 point to Defense is undervaluing shields significantly.
Using the sword by itself opens the possibility of grappling with your off-hand - after you or your opponent make a successful cover, you may have an opportunity to grab the pommel of their sword, lock up their arm in a key or a bind, or even get really close and throw them to the ground.
Successful attacks with the sword are seeking to place a thrust in the head or neck, or less optimally in the chest (as the ribs are much harder to get through). With cuts, you can strike a near-vertical stroke down, looking to hit the head, or the base of the neck. Horizontal strokes can be aimed at the head, neck, or arms (preferably the sword arm, as chopping that off will instantly win the fight).
Rising strikes are typically aimed at the hand or arm, as their reach and power is less (and it doesn't take much force to lop off a finger), but they can easily be turned into a thrust at the end.
Defences with the sword are ideally seeking to displace the opponent's blade and make a strike or thrust at the same time. In the broadest sense, same counters same. A downward blow from the right shoulder will deflect a downward blow from the right shoulder. A thrust to the face counters a thrust to the face. If this was all there is to it, swordfighting would be a piece of cake - obviously, there are many other options.
Fiore describes a "universal defence" using the sword in one hand - stand with your right side to the enemy, sword down and back pointing past your left foot. Against any sword attack, you can step offline and cut into the attack, deflecting it and positioning yourself in a great place for a counterstroke. Interestingly enough, the starting position for this is basically leaning away from your opponent with your sword in its scabbard.
If a defence ends up with the swords bound (i.e. stopped edge-to-edge), you have a number of options - wind your sword around theirs and strike, grab their hand or arm with your off-hand to control their sword and strike, disarm them, slam them in the face with your pommel or quillon, a kick in the knee or groin...