A forward contract is similar to a Future. Comparing it to a stock, is less than ideal since there are significant differences between the two... although you could very well write a forward on a stock price, this would be a specific instance of a forward and not the general rule.
The forward is like a future. It is a contract between 2 parties where one agrees to buy a given amount of an underlying asset at some point in the future, at a specific price that is set today.
But unlike the future, it is "custom made" and the amount of underlying asset represented by the contract, the expiration date, the margin requirement, the inclusion of a third party (clearing firm) that makes sure each of the other 2 keep to their word and the price are negotiable.
In a future contract, the only part that is negotiable is the price. Everything else is kept standard. Because of this, it is very easy to get in/out of a future contract, you buy it and sell it in the market where it is traded. The forward in the other hand is very iliquid and it is very difficult to get out of such a contract before expiration.
About your text book formulas.
e^(rt) is the formula for continuously compounded interest rate. It measures the value of money in time. The positive exponent brings the value into the future, while the negative exponent brings the value into the present (or past, depending on where you're standing)
The other part of the formula is the difference in price for the contract between the day when it is written and expiration date (on the second formula) or the spot price (in the first formula).
This is basically following the form: FutureValue = SpotPrice + CostOfCarry where CostOfCarry is simplified to the interest rate (this is true for a financial asset, not so much for physical commodities).
The FwdPrice is the price of the Fwd in the present taken into a future value adding the cost of carry.
The Fwd Value is the price of that Fwd in the future once the expiration price (K) is known, brought to present value.
At the time when the contract is written there is an exchange of money (margin), depending on how the contract is negotiated this can be the full price of the contract or a fraction of it.