The weapon categorisation between Simple and Martial is primarily a way of grouping weapons together for determining which classes have proficiency with those weapons. Proficiency with a weapon means that the character may add their Proficiency Bonus to the Attack roll.
Type
The type column in the above table distinguishes between Melee and Ranged weapons. Ranged weapons may be used at range. Melee weapons may be used to attack enemies within 5ft or may be thrown if they have the Thrown property. Attack and damage rolls with Melee weapons add the Strength ability modifier (unless Finesse, see below), whereas attack and damage rolls with Ranged weapons add the Dexterity ability modifier (see p176-177 of the Player's Handbook).
Ranged weapons attack at disadvantage if there's a hostile creature (that can see you and which isn't incapacitated) within 5ft (see p195 of the Player's Handbook).
Weapon
The name of the weapon
Cost (gp)
The cost of the weapon in gold pieces. Note that the table on p149 of the Player's Handbook expresses some costs in copper pieces and silver pieces. All costs have been converted to gold pieces in the above table to allow sorting by cost.
Damage
The damage dice rolled for the weapon. Note that were weapons have the Versatile property, two-handed use of the weapon will have higher damage.
Damage Avg
The average of the dice rolls for the weapon, to facilitate sorting the table by damage. Note that were weapons have the Versatile property, two-handed use of the weapon will have higher damage.
Damage Type
The Type of damage of the weapon. This is relevant as some Monsters have resistance to damage types. For example, the Peryton (see p251 of the Monster Manual) has resistance to bludgeoning, piercing and slashing damage (from non-magical weapons). REsistance means that the damage from the weapon is halved (see the Player's Handbook, p197), rounded down (see the Player's Handbook, p7).
Weight (lb)
The weight of the weapon in pounds. This attribute is used for encumbrance rules, see p176 of the Player's Handbook.
Light/Heavy
The Light property allows the weapon to be used in two-weapon fighting without the Dual Wielder feat (see p165 of the Player's Handbook for the Dual Wielder feat and p195 for the rules on Two-Weapon Fighting).
The Heavy property means that small creatures have disadvantage on attack rolls with such weapons.
Handed
Weapons are either one-handed, two-handed or versatile. As the descriptor implies, one-handed weapons can be used in one hand, allowing a shield (or another weapon) to be used in the other hand; two-handed weapons cannot be used in conjunction with another weapon or shield. A versatile weapon may be used either one-handed or two-handed, and has a higher damage roll when used two handed (denoted in brackets in the table above).
Thrown
A weapon with the Thrown property can be thrown at an enemy, the range of the throw is given in the Normal Range and Long Range columns. The same ability modifier is used for the thrown attack as for a melee attack, i.e. strength unless the weapon has the Finesse property. The ability modifier to use for Nets is Dexterity, as clarified by Mike Mearls.
The rules on recovering thrown weapons after the battle are not clear. Per p146 of the Player's Handbook, half of Ammunition can be recovered after the battle (if you take a minute to search). However, thrown weapons do not have the Ammunition attribute, so it's not clear that this rule applies. Thrown weapons are arguably less fragile than most ammunition, and is typically larger so harder to loose, so it would suggest recoverability is better than for ammunition (subject to the one minute rule). Common sense should also apply, for example if you throw a handaxe at an enemy on the other side of a river of lava, you will need to find away across that river of lava to collect the weapon.
Finesse
A Finesse weapon allows the player to make the choice of using the Strength or Dexterity ability modifiers for their attack or damage rolls. The same modifier must be used for both rolls.
Normal Range, Long Range
When attacking a target at up-to-normal range away, there is no attack penalty. Between the normal range and long range, attacks are made at disadvantage. Hitting a creature beyond the long range is not possible. Additionally, ranged weapons attack with disadvantage if there is a hostile creature within 5ft (regardless of whether that hostile creature is the attacker or not).
Reach
The Reach attribute allows targets to be attacked that are 10ft away (rather than the normal 5ft for melee weapons without the Reach attribute). This benefit also applies to Opportunity Attacks allowing the wielder of a Reach weapon to control a wider area of the battlefield.
Ammunition
A weapon with the Ammunition attribute requires ammunition in order to be used. For bows, arrows, for crossbows bolts, for slings, stones or bullets, for blowguns, darts. A character with such a weapon can only attack if it has ammunition available. Taking ammunition from its relevant container is part of the attack action.
Weapons with the Ammunition attribute can be used as improvised melee weapons.
Half of expended ammunition is retrieved at the end of a battle.
Loading
A weapon with the Loading attribute takes sufficient time to load that only one attack with this weapon is possible per action, bonus action or reaction. So, for example, a level 5 fighter with two attacks per action can only attack once if using a weapon with this attribute.
Special
There are two weapons with the Special attribute, the lance and net. The special rules are detailed on p148 of the Player's Handbook.
The lance is special as it is used at disadvantage if the target is within 5ft, and requires two hands to use when the wielder is not mounted.
The net is special in several ways, listed below:
The net does not do damage
A net cannot used to attack a creature that is Huge or larger, or a creature that is formless
A creature hit with a net becomes restrained (disdvantage on attack rolls and dexterity saving throws, attackers have advantage on attacks, see p292 of the Player's Handbook).
Getting free from the restraining effect of a net requires that the restrained creature, or an ally, spends an action and makes a successful DC 10 Strength check.
The net can also be destroyed with slashing damage, AC10, 5hp, freeing the restrained creaure
Similar to the Loading attribute, attacking with a net limits the attacker to one attack per action, bonus action or reaction.