It is certainly one of the groundcover types of Juniperus sp.
I haven't had to plant or specify this type of plant in so long, I no longer remember how to separate Juniperus horizontalis, Juniperus procumbens, J. conferta, J. sabina and all the selections and clones and probably trailing versions of normally shrubby or upright species. Regardless, you are experiencing the irritating effect of the juvenile foliage (the pointy sharp needles) on the plant you have. It may simply be an abrasive effect (like if you rubbed coarse sandpaper on your legs), or you may be allergic to it.
Since it is a woody plant, it will have branches that lead back to a trunk. If you want to eliminate it, you will have to cut all those branches off and work back till you find the trunks from which it all started. Then, either dig out that trunk/stump or treat it with a woody plant herbicide to ensure it is dead. I don't think these species resprout readily, but someone else may weigh in with additional experience.
Obviously, gloves/long sleeve shirt/long pants will be necessary to avoid contact with this plant's foliage to reduce chance of further irritation.
