Hopefully this isn't to complicated. I want to create a logo that "floats" over a piece of video. Ex; 1960 Batman with Adam West, occasionally the Batman logo would appear over the video to transition to the next scene. That type of logo is what I'm after. I would like to create a drawing and float it over the video, however the background of my image is constantly white. If I work on a smiley face for example how can I have only the smiley face appear and none of the background, zero white or any kind of color? Basically making the background of the image transparent?

Inkscape backgrounds are transparent by default. So just do your drawing, lol!
Even though you don't see the checkerboard pattern used by raster graphics to indicate transparency, it is indeed transparent. Although if you're converting to PNG, don't use the Cairo PNG. Instead use File menu > Export bitmap!

Also, if you go to your document properties (File->Document Properies) then, in the Page tab you will see a background "button". Click it. In the window that pops up make sure the A (called Alpha channel, the transparency channel) value is 0.