I am trying to set up an accounting program and would like to clearly understand what the terms "trading" and "non-trading" stock refer to.
Every company has Stocks.
For the stocks to be traded via some stock exchange, the companies must follow the eligibility criteria and guidelines. Once done, these are then listed on the stock exchange and can be traded. The advantage [amongst others] of listing is liquidity and stocks can easily be bought and sold.
Some small companies or closely held companies may not want to list on stock exchange and hence are not traded. This does not mean they can't be bought and sold, they can be outside of the market, however the deals are complex and every deal has to be worked out.
During the course of time a stock that is traded on a stock exchange, would either fail to meet the criteria or voluntarily choose not to be traded and follow the delisting process [either by stock exchange or by company]. After this the stocks are no longer traded on the exchange.
A non-trading stock or non-marketable security or unlisted security is one that does not trade continually on an exchange.
For tax purposes, this can mean a whole new ball of wax which I would prefer the experts address with an edit to this answer or a new answer.
For financial accounting purposes, this is when, say, one owns shares in an unlisted corporation and should be treated very carefully less one delude oneself.
For trading stock, the value can be known immediately by checking any valid data provider's price and marking to market. For non-trading stock, the value has to be "marked to model". This can get one into Enron sized trouble.
In this case, it's best to either leave the value of the stock at the purchase price and recognize gains upon sale, use a price from another honest transaction by third parties which are most likely difficult to attain, or to use some shorthand measure like applying the market P/E.
Be wary of strangely high figures for value from the purchase price by using a market average, and don't throw away the shares just yet if a strangely low one arises. This method can lead to strange results.