Angels are pure spiritual beings. That is, they are pure intellect with no physical bodies. However, they have the ability to somehow form a manifestation so that we can see them when it is necessary for them to communicate with us. Humans are composite beings. We share a physical nature with the animals, but we also have a spiritual and intellectual nature like the angels. So although we have some things in common with the angels, we are different from them because of our physical bodies.
When we die, our spiritual component is temporarily separated from our physical body, which usually undergoes decay. During this time, we are spirits without bodies, and in a way more like the angels. However, at the end of this age, at the resurrection, we will be reunited with our resurrected bodies, and will maintain this composite physical/spiritual form for eternity.
Note that even Jesus, who is God, has a physical body because he permanently united himself with human nature, and he has already undergone resurrection. He somehow has a physical body, although it is different in nature from the body he had before his death. Similarly, our future bodies will be of a different nature than what we have in this life.
Angels never had and never will have bodies, though.