The literal meaning of the noun tartan is: a woollen cloth woven in one of several patterns of coloured checks and intersecting lines, especially of a design associated with a particular Scottish clan.

This word has come to be also used allusively in reference to Scotland or the Scots—cf. Tartan army.

In particular, during the Troubles in Northern Ireland (i.e. the period of conflict in Northern Ireland from the late 1960s until the ‘Good Friday’ Agreement of 10^th April 1998), Tartan was used to denote a member of one of the Protestant street gangs, from their traditional support of Glasgow Rangers Football Club. This was explained by Simon Winchester, reporting from Belfast, County Antrim, Northern Ireland, in Army fears action by Protestants, published in The Guardian (Manchester, Lancashire, England) of Friday 17^th March 1972:

For the first time since the beginning of the present phase in the Ulster emergency, senior army officers are admitting to being seriously worried at the prospect of a steadily increasing number of Protestant-inspired violence.
[…]
One very senior officer said yesterday: “Our studies leave us to believe that there could almost certainly be some very horrible acts, inspired by Protestants, in some areas of Belfast and in the country. It should be stressed that there is no major or political or religious organisation behind either the attacks that have already taken place or those that we think might happen. What we are faced with are large numbers of politically motivated hooligans.”
[…]
The groups which are expected to perpetrate the first of these anti-Catholic attacks — taken either as revenge for IRA attacks on Protestant property or as open defiance of any political initiative which is interpreted as a loyalist sell-out — are the local Protestant street gangs, mainly known as “Tartans” because of their traditional association with the Rangers Football club.
Although all Protestant areas are known to have their Tartan groups the army at the Lisburn headquarters is particularly concerned at the prospect of attacks being made against Catholics in East Belfast, and in particular in the Ballymacarrett area, which lies between the river Lagan and the entrances to the Harland and Wolff shipbuilding yards.
For scores of years, Catholics in East Belfast have been overwhelmingly outnumbered by the local Protestants, most of whom work in the shipyards or in the local foundries. The Catholics live in two very small areas beside the river — the larger, known locally as Short Strand, contains about 800 crumbling houses and 3,000 people; the smaller, Willowfield enclave, has fewer than 100 houses, and only a few hundred Catholic inhabitants.

The Queen’s Own Highlanders were based in that area, and Simon Winchester quoted their Commanding Officer, Lieutenant-Colonel John Hopkinson, as using the term Tartan gang:

Lieutenant-Colonel John Hopkinson […] said yesterday that the Albertbridge Road, which is the interface between the Catholic and Protestant areas, had been the focus of most of these attacks. “We have about 10 or a dozen Tartan gangs in this area, mostly from the area of Ravenhill Road to the south. During the past three or four weeks these gangs have been coming up to the interface and stoning the Catholic houses. They have broken a few windows and hurt a few people — nothing very serious, but it’s a sign we can’t ignore.”
Local gangs include the Albert Tartans, the Young Newtons, and the Woodstock Tartans; the Pass Tartans, from the Donegal Pass across on the west side of the river, make occasional cooperative forays across the Albert Bridge.

In ‘Not bloody likely. I hate all of you’, published in the Evening Express (Aberdeen, Aberdeenshire, Scotland) of Friday 10^th December 1971, David Smith gave an account of his visit to the Queen’s Own Highlanders, who had just established their headquarters at Shortstrand Bus Depot, in Belfast:

Word had reached Shortstrand that the Protestant “Tartan Gang” had penetrated the Catholic area and was only 100 yards away, heading towards the depot.
Armed with pistols, Lt. Morrison and Sgt.-Major Duffus lead the way towards the Interface—the road dividing the Catholic houses from the Protestant community.
Troops in front checked for snipers around the Albert Bridge.
L.-Cpl. David Penman (24), of 4 High Street, Lossiemouth, said: “The Tartan Gang are a bunch of yobos aged between 18 and 26 who do night patrols in the Catholic area. They bash in windows and do hit-and-run raids.”
[…]
A tearful young mother showed the Queen’s Own the heavy metal bolts which the Tartan Gang, many of whom work in shipbuilding, had hurled through her windows.
Scores more bolts were found in the streets.

The earliest instance of Tartan gang that I have found is from the Belfast Telegraph (Belfast, County Antrim, Northern Ireland) of Monday 20^th September 1971. It illustrates the fact that Tartan gang (which is here opposed to Roman Catholics) denotes any Protestant street gang of young men in Northern Ireland—Portadown is a town in County Armagh, Northern Ireland:

Police stop Portadown gang fight

Groups of young men who were fighting each other in Portadown were dispersed by police on Saturday night.
The trouble began after some people had left a dance at the Pavilion in Chambers Park.
It is understood that one group is known as the Tartan gang and another were Roman Catholics from the “Tunnel” area of the town.
In one of the clashes a young man received an eye injury which required four stitches.
A police spokesman said they had succeeded in preventing any escalation of the disturbances.