Imagine a covalently closed circular double-stranded DNA plasmid. In its "natural", or "relaxed" state you could spread it out, flat, on a suitable surface.
Now imagine cutting one of the strands of the DNA (i.e., cutting the sugar-phosphate backbone, the way that an endonuclease would, leaving a nick, the 5'-end of the nick has the phosphate, and the 3'-end of the nick has an -OH group), and rotating that nicked strand in a clockwise direction, several times, around the other, intact, DNA strand. This does not involve breaking any of the basepairs, or unravelling the strands, you are just twisting it around.
Now imagine sealing that nick, by ligating the 5' phosphate to the 3' OH. At this point the dsDNA helix is slightly "overwound", and will no long lie completely flat. It has begun to be supercoiled; these are positive supercoils.
If, alternatively, you had rotated the nicked DNA strand several turns in the counter-clockwise direction before re-sealing it then that molecule would also start to be supercoiled, but it would be "underwound" with negative supercoils (and it would also no longer lie flat on the surface).
There is an enzyme called DNA topoisomerase I that cuts, and re-seals, one of the two strands like this. Another enzyme called DNA topoisomerase II cuts both strands before introducing supercoils.