When I see videos of the game, I see swords that are different from mine. Are there upgrades for swords and weapons in this game?
Yes, there are different weapons in Orcs Must Die that you can unlock either via playing the story or from purchasing the DLCs.
Also, once you unlock the Elemental Weaver you can buy upgrades for your weapons during each map.
The blade-staff is the only weapon in Orcs Must Die that can appear radically different in its appearance as the model used for the weapon in-game is tied to the current character skin used. There are currently two choices the standard blade-staff can be switched to; The hammer which is available to customers who have pre-ordered Orcs Must Die and the double blade-staff which became available after the community achieved 250 million collective orc kills. Selecting a skin will automatically change the appearance of the blade-staff. The blade-staff variations still function exactly the same as each other, it's simply a reskin.
Weapons can only be upgraded in-game by purchasing upgrades from the weavers which become permanently available after level 10 (Steel & Elemental) and 19 (Knowledge). Weapons cannot be upgraded using skulls. The skulls only upgrades traps.