What was the first machine that was able to carry on a conversation?

In 1986, the first PC therapist program was written by Joseph Weintraub. This program won the first Loebner Prize in 1991, and then again in 1992, 1993 and 1995.

In 1981 or 1982, Jabberwacky was founded, which is the foundation of the current Cleverbot. Jabberwacky appeared on the internet in 1997, reaching the third place for the Loebner Prize in 2003, the second place in 2004, and won in 2005 and 2006. In 2008, Cleverbot was launched as an variant of Jabberwacky.

I'm not sure these are really the earliest, but that also depends on what you want earliest (programming started, first conversation, first decent conversation, etc.). Also, it depends on what you call a "real response".

The well-known 'Eliza' program (Weizenbaum, ~1964) would appear to be the first.

Eliza was designed to model the emotionally-neutral response of a psychotherapist and this masks some of the weaknesses of its limited underlying pattern-matching mechanisms.