Algorithmic trading is the automated entry of buy and sell orders. This is sometimes used to mean all automated trading. It is also often used more narrowly to refer to automated trading that fulfils a manual buy or sell decision.
Complete automation of trading, including the buy or sell decision, covers a wide range of proprietary trading strategies, especially those that lend themselves to automation such as stat arb, and is essential for high frequency trading. It is also used for hedging.
Algorithmic trading can also refer to the use of automated systems to turn a buy or sell decision into a series of orders that is likely to get the best price. A simple example is turning simple buy or sell into an iceberg order. Algorithmic trading extends to much more elaborate automated processes: choosing trading venues (e.g. whether to send an order to an exchange or an ECN), deciding on the exact timing of order entry, etc.
This is the automation of a process that has long been (and still often is) carried out by human traders. The computers are cheaper and faster.