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The max number of weapons is 1 one handed and 2 two handed. Two handed weapons grant greater bonuses.

Swords: Swords add critical chance. The max critical chance for a unit is 30%. Heroes and units can only pass that cap with boosts (such as buffs and extras) and masteries (like the Blademaster’s Critical Edge ability which allows him to increase his max Critical Chance up to 40%). Critical strikes normally deal x2 damage, items, boosts and abilities (again the blademaster and other heroes) may allow increasing that damage (by using the engineering upgrade ability, which is kind of simple). Swords also grant defensive edge which reduces the attack speed of close enemies that attack by the front.

Daggers: Daggers deals backstab damage upon landing physical damage on the back or the rear of the enemy. Backstab damage comprehends on dealing 25% to 50% of the damage received by the target as bonus pure damage. Also adds a chance to make the enemy bleed.

Axes: Axes deal bonus damage depending on the missing life of the enemy. They also allow you to gain damage for each armor point of the enemy.

Staffs: Have a chance to negate a close frontal attack (parry). As an extra your attacks have a chance to make the enemy run away (this isn’t a fear state, just a trigger giving an order). One handed staffs grant less bonus.

Maces: Grants % chance bash (only half for range). Bash doesn’t deal bonus damage, unless masteries or extras allow it. Also increases damage dealt to structures (demolish).

Spears: Adds a chance on attack to pitch the unit, immobilizing it for 0.50 seconds (0.25 for one handed) and to deal bonus pure damage based on the enemy max life. Non-neutral units enter on a hold-position mode. Also grants beastslayer, which grants bonus damage against neutral units.

Halberds: They grant cleaving attack. Most of them are 2 handed.

Bows: They grant bonus damage against aerial enemies, bonus damage depending on the distance between the attacker and the target (% of the total stats) and Aim which adds a chance of dealing (x) bonus damage in a never-missing attack.

Fire-weapons: Grants extra attack speed depending on distance between target and attacker (more attack speed the far away the unit is). Also grants ammo attack which adds a chance to do a mini stun causing bonus damage. Increases attack speed against aerial enemies.
Claws: They add attack speed for each 1% of missing life of the enemy. Also adds a chance to Maim, slowing enemies AS and MS.

Whips: They have a chance to ensnare the enemy for a short duration on attacks, granting extra chance if the enemy is at range and u attack it by the back. Adds a chance on each attack to disarm the opponent or to slow it’s attack speed.
Wands: causes your hero to deal spell damage on attacks (x damage + a % of all stats and AP ability power). They add a chance of critical cast. Critical casts deal up to 2x times normal damage on single target spells and up to 50% bonus damage on AoE spells.

Rods: They add spell bash, wich adds a chance to stun when landing spells on enemies. It has a cooldown. Also grants force; adds a chance of knocking the enemy back (quickly), damage and distance increases with AP.
Scepters: Add spell vamp, wich causes the hero to regain life depending on spell damage dealt, only a 33% for AoE spells, max spellvamp is 40%, at exception of boosts and abilities. Also grants spell rime: causes you to gain AP for each 1% of max HP missing.

Spheres: Grants Magic Shatter, which makes your attacks deal 100% bonus damage as spell damage each X seconds, this also feeds on AP. Also grants Spell Penetration: allows your spells to ignore X quantity of Magic Resistance.

Spellbooks: They are really variable and grant various actively used abilities. Sometimes they come in the shape of scrolls.

Of course there will be items that may grant bonuses like these without necessarily being weapons or being the same class of weapons specified on here. Tell me if I forgot something.