"She gets letters demanding money."
Translation:Faigheann sí litreacha ag éileamh airgid.
Yep. "Airgead" is a noun - the word for "money". "Éileamh" is also a noun - what we call a "verbal noun", i.e. it describes the act of doing something. "Éileamh" means "an act of demanding".
If you've done the genitive case (an tuiseal ginideach), you'll know that combining two nouns to denote ownership involves changing one or more of them (usually all except the first one) from the nominative case to the genitive case. A good way to imagine it is "something of something" or "someone's something".
So, if you wanted to say "at a bag of money", you would put "airgead" into the genitive and it would become "ag mála airgid".
Here, we have "ag éileamh airgid". Just like with the example above, we have two nouns ("éileamh" and "airgead") put together in a sort of relationship of ownership. So, "ag éileamh airgid" would literally be "at an act of demanding of money".
Therefore, the direct English translation of the Irish here would be "she gets letters at an act of demanding of money".
All verbal nouns work like this, except in some circumstances.
Another example would be "I am playing music". "Seinn" is "to play (music)" and "seinm" is the verbal noun. "Ceol" is music. However, since we are putting a noun after the verbal noun, we must again put it into the genitive. So, it becomes "táim ag seinm ceoil".
Does that make sense?
With time and practice, you become able to predict the form, spelling and pronunciation of genitive forms. As a reference, focloir.ie has the genitive forms of nouns.