The Angels are initially portrayed as mysterious entities of varying powers and forms who are drawn to New Tokyo-3. They are the enemy of mankind, threatening to trigger the Third Impact. As the series progresses, it becomes evident that the Angels are drawn not to New Tokyo-3, but to what lies beneath, NERV Headquarters.
Although Adam and Lilith are classified as Angels by NERV and placed in the order of Angels encountered, they are actually the progenitors of life. Adam was found in the White Moon under Antarctica, while Lilith was found in the Black Moon under Japan. The two moons are seeds of life created by the First Ancestor Civilization.
The Angels that follow, including Sachiel, Shamshel, Ramiel, Gaghiel, Israfel, Sandalphon, Matarael, Sahaqiel, Ireul, Leliel, Bardiel, Zeruel, Arael, Armisael, and Tabris, are all believed to have been spawned from Adam, while humanity (the Lilim) was born of Lilith.
The goal of the Angels is to recover and/or be reunited with Adam. Humanity seeks to prevent this for two reasons; it would cause the Third Impact and eradicate what is left of mankind; and they seek to use Adam and Lilith for their own purposesthe Human Instrumentality Project. Ultimately the Angels and humanity have the same goal, to reunite with their progenitor.
This need to return to our progenitors of life is an allusion to the need for companionship that is prevalent throughout the story. It is the need for completion, the need to be one with God. Unfortunately even the Angels cannot coexist with each other, as mentioned by Misato in the End of Evangelion. This also goes a long way to explain why the Angels do not attack simultaneously.
The Angels are described by Misato as merely a different direction life may have taken, thus the Angels are essentially humans of varying forms. It is a fact that although the Angels are composed of different matter, their wave patterns are all similar to that of human DNA. And so if Angels are humans of varying forms, by the same token, humanity itself is but an Angel of a different form.
There is however a lack of consensus whether humanity can be classified as the Eighteenth Angel. Some people believe that because humanity was not spawned of Adam, this excludes it from being an Angelic life form. Others point out that Lilith herself is classified as an Angel, and thus her offspring, the Lilim (humanity), should likewise be classified as such.