Bands: Their relative importance in the early scene
Joy Division are not always thought of as being goth, despite being referred to as "gothic" at the time, but their influence on goth bands was considerable. Their sparse, haunting sound was quite unlike anything else around at the time and spawned a host of imitators, especially after Ian Curtis' death (Bauhaus' first album and the Sisters' first single were both slammed as being the work of poor Joy Division copyists, which was rather unfair on Bauhaus). Their use of minimalist and gothic art on record covers also had a lasting influence (for instance, the cover to the March Violets "Grooving in Green", designed by Andrew Eldritch, has a remarkable resemblance to the cover of "Closer").
Additionally, they may have been the source of the term "gothic" as applied to music.
Bauhaus are the first band who cannot be comfortably classified as anything other than goth. UK Decay and The Banshees could be considered punk, The Cure could be considered New Wave, Joy Division could be considered post-punk, but Bauhaus were unmistakably goth in music, looks, lyrics, art and style right from their first single. In many ways they were the archetypal goth band. They were also involved in the early goth "scene".
UK Decay are almost forgotten now, but they were important movers in the early goth scene. Abbo, from UK Decay, was responsible for using the word "gothic" to tag the emerging movement.
Siouxsie and the Banshees, unlike UK Decay, were not involved in the goth scene, but had a major influence on it in terms of both music and image. Their music had been called "gothic" as far back as 1979, and their music formed the template for a lot of female-fronted goth bands in much the same way that Siouxsie's looks provided the style for many female goths. Between them, Siouxsie and Bauhaus pretty much designed the early goth look.
And, once again, Siouxsie was important in establishing the use of the word "gothic".
The Cure, like the Banshees, were not part of the scene, and were less influential. However, their music and image fitted, and they were adopted into the goth canon. At least until they released Let's Go To Bed and a pop career beckoned...
The Sisters of Mercy, despite their later dominance of the goth scene, were not that important in the early scene. Their main claim to fame in the early years is being the first of the second wave of goth bands to release a single, though their first gig wasn't until several months later. It's odd that the Sisters came to dominate goth, since they were so different from the rest of the early goth bands: they had deep vocals and a drum machine whereas most of the early goth bands were characterised by tribal drumming, and none of them had vocals anything like Eldritch's. In 1982, the important bands were Bauhaus, UK Decay, the Banshees and Southern Death Cult: the Sisters had yet to emerge from the shadow of Joy Division (reviews regularly commented on their similarity).