Think of double-helix DNA as two sticky strings (say, one blue and one red) twisted together. To make a plasmid, take the twisted string and tie the ends of the red strings together, then tie the ends of the blue string together. You'll have a circle made up of two circular pieces of string that are twisted and stuck together.
Supercoiled plasmid occurs when the red and blue are twisted together so tightly that they start to kink up (Seriously, take a piece of string and twist it, you'll see what I mean.) If a plasmid is nicked, imagine untying the knot in the red string. Now it can untwist around from the blue string and release some of the kinks. It's still double-helical, since the two strings are stuck together (modeling the base pair bonding), but it's in an "open" or "relaxed" conformation.
Here's where my model breaks down: each of the strings has a direction, and the two strings' directions are opposite. That's the 5'/3' designation.
Enzymes usually recognize a palendromic DNA sequence (for example, EcoRI cuts 5'-G'AATTC-3'. Because it's palendromic, when you have the double stranded DNA, you have the same sequence reading either way:
(if you take your monitor and flip it upside down, you'll see that this sequence looks the same either way you look at it)
So, the enzyme recognizes the double-stranded sequence, and cuts each strand where I've put the tick mark, giving you a linear plasmid with the ends:
The string is still twisted and stuck together, but it's back to being a string rather than a circle.
Does this help at all? Or make any sense to you? Or was this not your question?