TP Bonus (found on weapons such as the Martial Knife, Martial Scythe, Martial Anelace, etc. along with the Shamo and Hagun) allows a weaponskill to use 100% additional TP from what you have when the skill is actually used. (Exception: Summoner's level 75 Blood Pacts earned with Merit Points give a TP Bonus effect of 40% (instead of 100) to the attack for each level learned beyond the first.)
For example, using Tachi: Yukikaze at 100% TP with a Hagun (Martial Great Katana) would have the same effect as though you used it at 200% TP with a non-martial weapon.
TP Bonus weapons are primarily used to increase damage on heavily TP-dependent damaging weaponskills (e.g. Shark Bite, the final three Tachi (Great Katana) skills, Steel Cyclone, Savage Blade, Spiral Hell, and most of the other high-level weaponskills). It can be used on weaponskills with other modifiers (Asuran Fists for example has a TP modifier of more accuracy, Viper Bite has a TP modifier of more poison) but the extra damage from a damage TP modifier is the common use.
It does not allow the user to gain TP any faster, that's what the "Store TP" Job Trait and effect on certain items does.
TP bonus only works if the weapon is equipped in the main hand; a TP bonus weapon in the off hand has no effect, and a TP bonus ranged weapon will not affect a melee weapon and vice-versa.