The most common and most likely answer to this crossword clue is the 6 plus 5 letter word SCREEN SAVER
I've seen this crossword clue in The Daily Telegraph
S
C
R
E
E
N
S
A
V
E
R
More clues leading to the result SCREEN SAVER
Buddy explains!
The straight part
We know that "something that might appear on monitor" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case something that might appear on monitor = screen saver.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.