Market Price: "Market price" for an order means the lowest current ask price (for buy orders), or the highest current bid price (for sell orders). Market price is essentially the best offer on the order book, which will be different for buyers and sellers, since the best offer for buyers is the lowest sell order on the book and the best offer for sellers is the highest buy offer on the book. Market price is important for understanding how stop or take profit orders trigger. A stop loss sell order, for example, will trigger when market price (the highest current bid since it's a sell order) is less than or equal to stop price.
Best Average Market Price: The "best average market price" for a market order means the (weighted) average price of the best current asks or bids that can fill the order. This is important for understanding how market orders fill. Typically a market order will be filled by several opposing orders on the book. A market buy order will be filled by several of the lowest asks on the book and a market sell order will be filled by several of the highest bids on the book. Hence, the average fill price of a market order will be the best average market price - a weighted average of the different orders at different prices that filled the market order (weighted according to the size of the filling orders).
Position: Even though only currency pairs are officially considered "positions," currency exchange will also be described below in terms of opening or closing a position (for those who actively exchange, this will probably seem natural).