Native resolution is the hardware's number of pixels available for display. Ie, a 1024x768 LCD will have 1024x768 actual pixels for display. In such a display, running a higher or lower resolution requires resampling of the data, so that it can be scaled up or down for the actual number of pixels present. (Upscaling is not often done, except on some LCD TVs and LCD Rear projection sets.)
For this reason, it's best to always run at the native res with an LCD... otherwise your picture can start to look funny or distorted.
Native vertical resolution is simply the number of horizontal-lines available on the display (768 in my example above). In other words, the number of lines stacked vertically that comprises the image. A TV with 720 native vertical lines can display 720i or 720p best, as this is a matched resolution. Higher and lower resolutions will need to be resampled.
Put another way, if you feed a 1080 image to such a TV, you are downscaling the data to 720, since your TV can't ever show 1080... hence this is very important if you are picking an LCD or LCD RP TV. Similarly, feeding a 480 signal to this TV would require upscaling to 720, which is not an "even" number, as you can see... and can result in some distortion.
So, when a TV says it's 1080, 720, and 480 capable, you need to read the fine print to know what it's REALLY doing. (Early LCD RPs would say 1080 in big bold print, while they ran a 480 LCD inside... so you never got true HD!)
EDIT: Mixed up vertical lines of res vs horizontal lines of res - corrected. Added last paragraph.