Margaret Thatcher (1926 - 2013) was the Prime Minister of Britain between the years 1979 and 1990 and is by far and away the most memorable character of Spitting Image. She is also one of the most frequently-appearing characters, making some form of cameo in virtually every episode.
As the staff of the show were predominantly left-wing, the show took a very anti-Thatcher stance. Thatcher's puppet was remade every series to make her look more evil. She was portrayed as an outright fascist, going so far as to take political advice from her next-door neighbour who bore a frightening resemblance to Adolf Hitler. By the second series Maggie was dressed in a suit and smoking a cigar, reminiscent of Winston Churchill. The idea behind this was suggested by Anthony Asbury, who thought making Thatcher dress like a man would highlight just how evil and insane she was.
Indeed, as the series progressed, Thatcher becomes more and more unstable. In the first series is merely treated as a bullying woman who insults her Cabinet. As the show continued she began physically abusing her staff (one specific seat in Number 10 leads to a crocodile pit, to whom she fed Chancellor of the Exchequer Nigel Lawson) and growing more and more attached to Ronald Reagan, who was being shown as a stir-crazy war-hungry old coot from the first episode. Some sketches even hint towards a romantic relationship between the two world leaders.
Perhaps another aspect of the show that made Thatcher so memorable is the fact that, for the show's entire duration, she was voiced by Steve Nallon, who has achieved fame outside of Spitting Image thanks to his fantastic range of impressions. Indeed he is still hired today for private shows to do impressions of his Spitting Image characters. He has even appeared under the guise of Thatcher himself in many television shows.