What is the Damascus road experience?

The two apostles on the road to Damascus were Cleopas and probably Luke, since it was he who wrote of the account.
The Lord was showing them that He was indeed resurrected even though they did not recognize him at the time.
He appeared to many after his resurrection including many women, Mary mother of Joses, Joanna, Salomi mother of James and John

This is based on Saul's journey to Damascus where he was traveling to further persecute the Christian converts there. On route, the Lord appeared to him as a brilliant light which caused Saul to fall before it. Jesus spoke to him, and Saul was totally transformed from a vehement and vindictive unbeliever to a humbled man seeking the Lord's will.

In modern day usage, to refer to something as a ' Damascus road experience' is to refer to a profound life-changing experience, that turned one from skepticism to belief in the Jesus whom one previously mocked and rejected.