Design your logo in your image editor. Match the file format to the software--perhaps PNG for Final Cut Pro, or JPG for others. Create an image that gets the point across, but still manages to be eye-catching and capture the spirit of your show or network.
Remove the background content from the image file--basically everything that you don't want to appear on screen. Use the magic wand function if your image editor allows it, or manually erase the content while being careful not to erase parts of your logo.
Save the file, close out of your image editor and open up your video editor. Import the logo as source material alongside your video clips. Lay the image onto your timeline where you would like it to appear and for how long. Make any other adjustments you wish, such as for size, transparency or motion of the graphic.