Channel packing means using different grayscale images in each of a texture's image channels... Red, Green, Blue, and optionally Alpha.
These channels are usually used to represent traditional RGB color data, plus Alpha transparency. However each channel is really just a grayscale image, so different types of image data can be stored in them.
Individual channels can be extracted by a shader to use them for particular effects, for example the red channel for glow, green channel for specular, blue channel for sound types, alpha channel for physics info, etc. Each channel can have a totally different layout, and thus use different UVs.
Swizzling means to change the order of the color channels in a texture. For example, moving the red channel into the alpha channel, to prepare an image for DXT5nm compression.
Usually swizzling is done to create better results when using image compression, since the green channel sometimes uses more bits. This is because the human visual system is biased to perceive more values in the green range than in red or blue.