The term "tartan" is indeed the correct term for the weaving of alternating colors in the warp and weft to form square patterns.
However, "kilt" also has a very specific meaning - the modern Scottish male garment. The kilt was invented by a tailor in Lochaber (a district of Scotland) in the early 19th century. The tartan fabric is sewed into permanent pleats with a waistband and lining, laps over in the front and is fastened with three leather straps and sometimes snaps. The kilt pin at the bottom of the 'apron' is ornamental only and never fastens the layers of cloth together.
Before that time, there was a long period when wearing tartan was banned by the government (post-1745 rebellion). Prior to 1745, tartan was worn as a "belted plaid" -- essentially a huge unmodified rectangular piece of tartan cloth approximately 1 1/2 yards wide and anywhere up to 10 yards long. When ol' Lachlan Dubh got up in the morning, he would shake out his plaid (it was also used as a sleeping bag) and lay it out on the ground. He pleated the whole mess by hand to reduce the length to manageable proportions, then slipped his belt under the cloth. He'd then lie down so that his knees were about level with the bottom selvage, flip the two ends across in front, buckle his belt, and stand up (panting slightly). At this point the plaid is kilted around his waist, and the surplus cloth is hanging down behind like a tail. He could leave it hanging there or pin it to his shoulder, or if it was raining turn it over his head.
Obviously this was a heck of a lot of trouble for the modern Scot to fuss with, hence the invention of the kilt. When the kilt was invented, the upper half that was cut off became a separate garment known as a "plaid". You can wear that over your shoulder, but about the only place you'll see it now is on pipers in the Army, in full and absolutely correct day wear (in which case it's very small and just worn folded in a rectangle over the left shoulder), or in ultra-formal evening dress with the "Prince Charlie coatee" instead of black or white tie.
You'll still see the belted plaid worn by the re-enactment types and some historical sticklers at Highland games. It's hard to find cloth woven wide enough for men of modern height, though - 60 inches is about the widest you can find, and that is going to be a bit skimpy either above or below. :-D