In RO2, your weapon is personal. You get to choose it's name while in character creation (you can't change it later), and it can't be traded, sold or disposed. You can, though, get a second weapon (read the Getting Second Weapon guide for more information about this) or change it's name subfix at the blacksmith.
Blacksmiths can change your weapon type, too. You can have a Long Sword, Short Sword, Sword Stick or Dampf Finte (Rifle). Each weapon is recommended for a certain class, since most of the class skills will ask for this certain weapon to work. Long Sword is for Swordman, Short Sword for Thief or Clown, Sword Stick for Enchanter and Dampf Finte for Recruit.
You can open the Weapon Equipment Window pressing U while in game. This window shows how many weapons do you have, their level, auges, alter stones and other information. As you use your weapon in battles, it will gain experience and level up. You can see your weapon experience in the Weapon Equipment Window. Once your weapon has leveled up (a light and an icon will appear) you must take your weapon to a blacksmith. The more level the weapon has, the more powerful it'll be.
Also, you're able to change the weapon appearance through the use of Auges. An Auge is a sphere stone, similar to a cat's eye, that is dropped by monsters. Each Auge has a required weapon type and required level for it's use. Just double click them and select the weapon you'll like to change.