Far as I recall, bein' from this side o' the water and all, a serial was something made up of maybe six or eight episodes but
self-contained, never intended to have a sequel. So "Children of the stones" --- brrr! "A very British coup" and "Edge of darkness" were serials, ie they never went beyond the one "season", as you guys would say, but something like "Doctor Who", "The Tomorrow People" or, er, "Black Beauty" was a series, as it moved from first to second, and so on.
Seasons are arranged, it seems, in a certain order: they begin in one month and end in another. Nowadays we buy most of our stuff from you guys, so we get seasons, but anything home-produced (RTE or BBC or ITV etc) is still referred to as a series. It doesn't have to run over any particular span of months, unlike US stuff, which, well, does.
Just to confuse everyone though, RTE here still refer to their "new season" for any upcoming shows due in the next few months.
But yeah, that's basically it. Jansz would be right in terming a serial a mini-series; more or less the same thing, except perhaps a mini-series might be picked up and made into a full series (thus acquiring seasons) whereas a serial would not. It would be self-contained, with usually no scope for any continuation. The story would finish in the last episode, and there would generally be no loose ends to be tied up.
Any clearer? No? Sod ya then!