A spot rate is applicable to a transaction that is taking place immediately, whereas a forward rate is a rate that is agreed for some time in the future. According to Investopedia, forward rates are based on the spot rate and are then adjusted for the cost of carry.
For example, if an exporter has an order that needs to go to Europe immediately, he or she would have to trade U.S. dollars for Euros using a spot rate. Typically this sort of transaction is completed within two days.
If that same exporter had an order that he or she knew would not be ready for six months, they could agree the transaction based on a forward rate. That means the rate will be agreed now, but the transaction will not take place for six months. There is an obligation on both parties to honor the agreement, regardless of the state of the order or what is happening in the currency markets. This allows the exporter to engage in hedging so that they are not as affected by negative swings in the currency market.
Spot and forward rates are common in currency markets but they are also used elsewhere, including in the trading of bonds and commodities.