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The 2nd measure of the Hungarian Rhapsody no.12 has the word "trem." What does that mean?

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The trem. means tremolo which means to alternate between two beamed notes at the speed of the beaming. If you look at the beams in the score you'll see they are at 32nd note speed hence the 3 beams. The reason for the trem. marking is there because you may just mistake the notes written for 32nd notes instead of a tremolo.

Hungarian Rhapsody no. 12

I suggest listening to the piece to get a feel for how the tremolos should sound.

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RIGHT HAND PLAYS A# AND C#, rapidly going from one to the other, sounding like a tremolo effect, lasting the whole bar. It's called tremulando.The left hand alternates between E and Fx (F##). Giving a tremulous low diminished seventh chord.

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    Not sure where you're seeing E double sharp - the left hand in the Schlesinger first edition, the von Bülow, Paderewski and von Sauer editions all show the left hand bouncing between F double sharp and E natural, which makes sense for a diminished seventh chord. – user16935 Apr 18 '16 at 3:55

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