What has already been said is quite correct that Finney popularized the phenomenon which became the altar call by initiating the 'anxious' or 'mourners' bench.' Asahel Nettleton opposed this technique. D.L. Moody later employed the 'inquiry room'; Billy Sunday's approach is known as the 'sawdust trail'; and Billy Graham really made the 'altar call' famous.
For some further study, probably the best book on the subject is The Altar Call: Its Origins and Present Usage (2000) by David M. Bennett.