Tartan was originally the term used for the way the cloth was woven. It actually had nothing to do with a plaid pattern at all. (Incidentally, plaid originally meant "blanket" and had nothing to do with a design or pattern either).
Take for example, a standard weave, which looks like this:
Under One, Over One, Under One.....
The next row:
Under One, Over One, Under One....And so on.
Take a look at your bed sheets. If they are cotton or a cotton blend, they are most likely a standard weave.
Tartan, which we now call "twill", is woven like this:
Over Two, Under Two, Over Two, Under Two...
Under One, Over Two, Under Two, Over Two, Under Two...
Under Two, Over Two, Under Two, Over Two...
Over One, Under Two, Over Two, Under Two...etc.
This creates a diagonal effect within the pattern itself. Look at your jeans (Levis), denim is a fairly complex kind of twill. Notice how there seems to be diagonal stripes going through the fabric. This appearance happens even if the stripes of the actual pattern are at perfect 90 degree angles.
The oldest piece of tartan dates to the 3rd century AD. Found near Falkirk, in an earthenware pot, it was covering about 1900 silver Roman coins. It had a checked pattern of undyed wool which consisted of dark brown and light brownish green, two common colors of the native Soay Sheep.
Tartan, to the ancient Celts, was first a form of fashion and later a symbol of wealth and power. The more stripe and colors, the more powerful and/or wealthy a person was. Therefore a simple farmer may have a tartan with only a few colors and large checks, and the Queen or the Clan Chief may have a pattern with 6 colors and an intricate pattern sett.
Clan affiliations did not appear untill much later. At this point, clan affiliations were shown in other ways: like putting a ribbon or feather of a particular color in your cap. "Yankee Doodle went to London....."
By the mid-1600s district patterns started to crop up; although, this may have occured more because of the local weaver than anything else. Dye was expensive and labor intensive. A weaver may have a large quantity of red and blue dye in stock "and if you want green its gonna cost ya extra". So, a whole lot of red and blue tartan would be seen in one area or another.
One item of note here is that, most areas were populated by a family or a few families. So the district tartans were, in that respect, clan tartans by default.
Another item of note is that, if you are going to wear a tartan for re-enacment purposes you do not have to worry about your family lineage or offending someone for wearing a pattern that is not "yours". You may want to look into the district patterns, especially if you are portraying a later time period; however, it is most likely not necessary.
After the Jacobites (Scots loyal to the Scottish crown that was usurped in 1688) rebelled at the Battle of Culloden in 1746, the English banned the wearing of all tartan. This ban was repealed in 1782 and the Scots had a widespread hope to re-establish the identity of Scotsmen as Scotsman (and not English or British). They were helped out by Sir Walter Scot in the early 1820s, as well as King George IV, who threw great parties where no one was admitted unless in "true highland dress". Each clan chief was expected to identify and validate a pattern for their clan. With this, the concept of the "clan" tartan was strongly set in place. Queen Victoria, who had a great liking for all things Scottish, only encouraged it all. It soon became a matter of pride to have a tartan all of one's own. The tartan, therefore, proceeded to become the National dress of Scotland.
Today there are over 2500 registered patterns as well as hundreds of non-registered and private variations, encompassing Scotland, Ireland, Canada, the US and even Australia!