They can read your last 8 journeys, your PAY-G balance, when you last topped up (and how much), view the four "slots" on your card, (for Zonal tavelcards / bus passes, including those which have expired, except where they have been overwritten as all the other slots have been taken). They can also be used to view if you have a discount entitlement (16-25, for example), and also if you have loaded an OEP.
They can also be set up to know where they/you are, (NR, LUL/DLR, Bus), (Z1, Z2, Z3 etc, Z12, Z34, Z8 etc, or "No Zone"), and use this to determine whether or not your ticket is valid in that location at that time... although this function is not as reliable now, as things have become more complicated. For example, on a bus a red light may be displayed, (you have failed the inspection), as you have not tapped in recently... however the inspector needs to tab through your card details to see if you have a travelcard/bus pass!
They cannot in any way shape or form add or deduct money from your card