Would someone be so kind as to explain the 3/10 year ban?

I will do my best and I would get it confirmed as it is only my understanding, which may be slightly misguided. If you have unlawful presence in the USA for plus 180 days (this can be caused by an expired visa, illegal entry etc..) then you are subject to the 3/10 year ban. Not quite sure who they determine if it is 3, 5, 7, 10 years......one would think that it is dependant on how long over the 180 days you are, but it wouldn't surprise me too much if it was at the INS' discreation.

There are two separate penalties for unlawful presence: 1) more than 180 days but less than one year of unlawful presence makes that violator inadmissable to the US for THREE years
2) more than 365 days of unlawful presence makes the violator inadmissable for TEN years; there is no "in-between" penalties.
The unlawful presence must have been continuous, not cumulative.
There are waivers available for spouses/children of Amcits and LPRS, and also potentially available for non immigrant visa applicants. The latter are far more discretionary - a national interest is generally required for non immgirant waivers.