The M2 is the main road north from Belfast, carrying traffic from the city to the north and north-west of the province. Thanks to the geography of the city - constrained by the mouth of the Lagan to the east and the steep hills to the west - all traffic heading even vaguely in this direction is concentrated on the M2, which splits and branches several times as it gets further out. At the Belfast end, there are five lanes in each direction, but spurs and side-roads take their share of the traffic and before long it's a two-lane motorway in open countryside.
Most people in Northern Ireland are unwilling to allow road numbers to interfere with their driving. As a result, the M1 is "the motorway" and the M2, having been completed a year or two later in 1966, is "the new motorway" (though really, it depends who you ask: some people deny this completely).
The "new" motorway is central to NI's motorway network (what there is of it). There are only two motorways it doesn't encounter. Of the ones it does, two are direct continuations and the other two are entirely undeserving of their own numbers.
The M2 is now permanently split in two: filling the gap between its two sections is the A26, which has been upgraded and dualled, making any M2 completion plans redundant. Don't hold your breath for the final section to its intended destination of Coleraine either.
Some of the M2's junctions are worth closer inspection. J2 is now a fearsome knot of sliproads but began life as a simple fork, while J1b is the only motorway junctions in the whole of the UK where you can exit from the motorway, but once you've done so, you can't get back on. Junction 7 used to be the same way, but the missing entry sliproads were provided in 2008.