Standard definition television ("NTSC") is 480i, a far cry from HDTV's 720p and 1080i. The "i" stands for interlaced. This means that every 60th of a second, half the image is shown. The first 60th shows the odd lines of the image, the second 60th shows all the even lines. Your eye/brain combines these into one image. Most CRT "tube" TVs we all grew up with are 480i.
The main problem with 480i is that it isn't a lot of resolution, so as TV sizes went up, the perceived picture quality went down. Hence the need for HDTV.
DVDs are technically 480i, even though all modern players can output these as 480p or higher.
It's important to note that HD is significantly higher in resolution, and even on smaller screens this difference will be noticeable.