I'm not sure if your question was about the English sentence or the German, so I'll try to answer both. "On" literally means "on top of." So if you say "The woman is writing on the desk," you are saying that she is making marks on the table's surface (Die Frau schreibt auf dem Tisch). Similarly, "Die Frau sitzt am Tisch," should be, "The woman is sitting (or sits) at the table." "The woman is sitting on the table," is "Die Frau sitzt auf dem Tisch."
"Am" is a contraction of "an dem", which does not usually translate to "on" when talking about physical space, but prepositions are generally tricky.
Lost_in_light, I'd like to thank you for such a simple yet great explanation. I never thought I'd learn so much English whilst studying German than I have from comments like yours and many other's. As a non-native speaker I've struggled for years to understand some its subtleties and I'd like to encourage you to keep up the good work. Shönnes Dank!
Lost-in-light gives a good answer; however, it should be noted that sometimes the German "an" can mean "on" in physical space.
When speaking literally of "on" in physical space, "auf" usually means on top of (i.e., it is vertically over it), whereas the German "an" is more like alongside or horizontally "on top of" (actually next to) - see the picture on the wall example below.
Examples: Ich bin am Strand. = I am at the beach. (I'm on the beach, but next to the water's edge.) Compare "Ich bin am See" (I'm at the lake, nearby it) and "Ich bin auf dem See" (I'm literally on top of the lake).
Das Bild hängt an der Wand. = The picture is hanging on the wall.
Sie steht auf dem Berg. = She stands on the mountain. (She is on the mountain's peak.) Sie steht am Berg. = She stands on the mountain. (She is on the side of the mountain.)