"The woman is writing at the table."
Translation:Die Frau schreibt am Tisch.
"am" is a contraction of "an dem" = "at the".
If you say "an Tisch", you've left out the article, which is incorrect; in "am Tisch", the article is kind of invisible, but still included.
For much the same reason why it's not "The woman is writing on the table".
I think "auf" isn't actually wrong if you want to focus on the act of writing = the fact that the pen touches the paper on the table (not e.g. on the woman's lap), but it's much more elegant to use "am Tisch / at the table", focussing on the place where the woman is sitting while she does the writing.