Moortje was her cat, who she had leave behind. She said this cat was female in a letter but others say male. They first got Moortje in early 1942. Per instructions they left behind, the family of a friend of Anne's (referred to as "Toosje K." in FAF) took in this cat after the Franks disappeared into hiding.
"I miss Moortje every moment of the day and no one knows how often I think of her; whenever I think of her I get tears in my eyes. Moortje is such a darling and I love her so much, I dream up all sorts of plans in which she comes back again..." Anne wrote in her diary on 12 July 1942.
Mouschi was Peter's cat, a male. Miep Gies described him as a "spritely, lean black tomcat, very, very friendly." It seems Miep looked after this cat after the raid as a second office cat and then the office cleaner took him in at her home.
Moffie was the warehouse cat, another male. Miep Gies described him
as a "big, fat black-and-white tomcat with a slightly battered face." In
English translations (and probably French) of the diary, this cat is called
Boche. This is a throwback to the first translations trying to find
an adequate translation for Moffie. (Boche is
what the French called the Germans in WWI.) The problem has not
been corrected to this day. Anne never called the cat Boche.
Anne explained (in her diary: 12 March 1943) that the cat was named for the Germans some time earlier because he picked fights with Tommy, a cat named for the good stalwarts, the British. (Tommy had left the scene by the time they went into hiding.) Miep explained about the name Moffie in her book: "A Moffen was a biscuit in the shape of a fat little pig. our cat was known to steal food from other houses in the neighborhood, just as the Germans were doing with our food." During the war, the Dutch were also referring to the Germans as "Moffen." The cat's being a bit ugly did not help his prospects for a sweet nickname. The singular for Moffen is a Mof, and it seems that Moffie was an affectionate or fun variation on Mof.