Good question.
I will start: the white line is not evident in a shod horse because it is on the sole and the shoe will cover it.
You need to understand the structure of the hoof http://www.ironfreehoof.com/horse-hoof-anatomy.html
A tight white line is desirable.
A stretched white line is weak, prone to disease or indicative of disease, and to gravel ingress in an unshod horse.
The white line is the visible separation between the insensitive outer horn and the sensitive living part of the foot. It's slightly elastic and helps cushion the inner structures of the hoof and to protect against external contamination. Although it actually runs through the whole hoof you can only see it on the sole when the foot is trimmed. You can see it on a barefoot horse of course - and on a shod horse the farrier will use it as a guide to where the nails go - anything that goes through the white line is a no no.
Horses can suffer something called "White line disease" where this structure softens or breaks down allowing bacteria to enter the foot. One of the common names for white line disease is seedy toe - you might have heard that more often. If the white line is healthy and tight the hoof is protected both from bacteria etc but also to a normal extent from concussion. If the line is broken, weakened or damaged then the hoof will need urgent supportive care.
