"The woman is eating at the table."
Translation:Die Frau isst am Tisch.
I suppose, "beim" means "around the table (somewhere near / doing something that doesn't need a table, but the table is there)", "am" is actually there (eating there or playing cards). This example shows it perfectly: "The woman is eating at the table.". She is using the table for eating her lunch or breakfast or dinner. This sentence shows that the woman is using the table not the chair or the bed. Any confirm? :)
Yup, that's pretty much it. It was already explained elsewhere in this thread.
"Beim" in this case would actually be the contraction of "bei" and "dem". So, breaking it up into "Die Frau isst bei dem Tisch" handily translates to "The woman is eating by the table."
Likewise, "am" is a contraction of "an" and "dem", so again, you can rather easily, almost word for word, translate "Die Frau isst an dem Tisch" to "The woman is eating at the table."
That's a little bit of a tricky one. "Am" translates to "at". "Beim" means "near" or "nearby".
So, the sentence "Die Frau isst am Tisch." definitely means that the woman is eating at the table, i.e. she is sitting at the table, whereas "Die Frau isst beim Tisch" would translate to "The woman eats near the table", i.e. she is not sitting at the table, but she is near it, whether standing or sitting or crouching or kneeling.