yeah... fields are a bit hard to get your head round at first. Like many things it started off to do with technical limitations in TV.
Did you know that US uses 30fps and Europe uses 25fps because originally it was cheaper and easier to control the picture scan rate from the alternating current frequency (US uses 60hz electricity and europe 50hz)? Also (if I remember right) the 29.97fps thing is because when US introduced color TV, they found that the color subcarrier was interfering with the FM sound, so they had to change the frequency a little bit. (they could do that more easily by that time).
Lots of these standards remain because TV has always had to be backwards compatible...color had to be backwards compatible with black and white (why TVs work with YUV not RGB) etc. There's loads of interesting stuff in film and TV history. Nowadays, new digital standards are being made. No doubt we will one day laugh at MPEG and H.264 and how we had to compress images back then!
Fields is one of the hangovers. Basically, fields squeezes two 'half height' pictures into every frame. Each field is a picture, separated by 1/60th of a second, but the two pictures are interlaced. That is, line 1 shows picture 1, line 2 shows picture 2, line 3 shows picture 1, line 4 shows picture 2 etc.
When an old cathode ray tube TV scanned the screen, it would first scan the information of all the odd lines, then the information of all the even lines, with 1/60th of a second gap. For every 'frame' of infomation, the TV scans twice - giving an effective frame rate of 60 frames per second. Cunning stuff.
If you interpret the fields the wrong way round, there's the danger that instead of 1,2,3,4,5,6,7,8, the picture order will go like this... 2,1,4,3,6,5,8,7 etc. Most unnerving to the eye, I assure you.
You can emulate the effects and feel of interlaced footage on your desktop by creating a composition that runs at double your interlaced footage's frame rate and then time stretching the footage 200%. That's how I usually double check that I've got the fields interpreted the right way round.
These days, progressive images are often much preferred.... not least because they feel much more filmic. Interlaced footage tends to feel much more video-like. A bit like sport or TV news.
'Scuse the rambling answer, but it's quite a complicated thing, y'know.