Having the image or likeness of God means, in the simplest terms, that we were made to resemble God. Adam did not resemble God in the sense of God’s having flesh and blood. Scripture says that "God is spirit" according to John 4:24 and therefore exists without a body. However, Adam’s body did mirror the life of God insofar as it was created in perfect health and was not subject to death.
The image of God refers to the immaterial part of humanity. It sets human beings apart from the animal world, fits them for the dominion God intended them to have over the earth, see Genesis 1:28, and enables them to commune with their Maker. It is a likeness mentally, morally, and socially.
Mentally, humanity was created as a rational, volitional agent. In other words, human beings can reason and choose. This is a reflection of God’s intellect and freedom.
Morally, humanity was created in righteousness and perfect innocence, a reflection of God’s holiness. God saw all He had made (humanity included) and called it very good, Genesis 1:31.
Socially, humanity was created for fellowship. This reflects God's triune nature and His love. In Eden, humanity’s primary relationship was with God. Genesis 3:8 implies fellowship with God, and God made the first woman because "it is not good for the man to be alone", Genesis 2:18.
Part of being made in God’s image is that Adam had the capacity to make free choices. Although they were given a righteous nature, Adam and Eve made an evil choice to rebel against their Creator. In so doing, they marred the image of God within themselves, and passed that damaged likeness on to all of their descendants, Romans 5:12. Today, we still bear the image of God according to James 3:9, but we also bear the scars of sin. Mentally, morally, socially, and physically, we show the effects of sin.
The good news is that when God redeems an individual, He begins to restore the original image of God, creating a "new self, created to be like God in true righteousness and holiness", Ephesians 4:24. That redemption is only available by God’s grace through faith in Jesus Christ as our Savior from the sin that separates us from God, Ephesians 2:8-9. Through Christ, we are made new creations in the likeness of God, 2 Corinthians 5:17.