The Writings offer a tremendous amount of material on angels. The book "Heaven and Hell" offers detailed discussions as it describes heaven; "Conjugial Love" has much to say about marriage and romantic love in heaven; "Divine Love and Wisdom" offers insight into how angels in their nature reflect the nature of the Lord. So we'll offer some basics here and recommend those books to those who want more detail.
(References: Divine Love and Wisdom 231; Divine Providence 60-67)
Basically, the Writings say that if people in this life open themselves to the Lord, follow the Lord's teachings and let the Lord change their selfish desires into generous loves, they will go to heaven as angels after they die. If they don't, and instead embrace their selfishness, they will go to hell as evil spirits. The Writings also say that this is the only source of angels and evil spirits - they were all once people. There is no separately created race of angels, no fallen angel Lucifer who is now the Devil; that belief is based solely on a few lines of misinterpreted scripture.
This makes sense if you look at it logically. If the Lord could create beings that would live in love and harmony with him with no possibility of evil, why would He have bothered with us? Why not just make more of them? The fact is, such beings would not have any choice in their actions, making them no better than animals. And ultimately, if they were purely good then they would really just be extensions of the Lord, so in loving them He would be loving Himself. The reverse is true of the idea of Satan or "the" Devil. The Lord creates us from love so that he can love us, bring us to heaven and make us happy. For Satan to exist, the Lord would have had to create him, and it would be contrary to His essence to create something that was not intended for heaven, for joy, and for union with the Lord.
So angels were once people, who got to be angels by embracing the idea of being good and followed the Lord's teachings as best they could. The Writings make it clear these people can come from anywhere, from any religious background. Some churches may have doctrine that is closer to the truth than others, but the point of any religion is for people to desire to be good and try to be good using the tools they have.
When those people die, they go first to a place called the "world of spirits." There everyone who has recently died can learn about the Lord and spiritual life and prepare for heaven. There also, people's inner affections start showing on the surface; those who are ultimately evil start losing the ability to cover it up, and the love starts shining through for those who are ultimately good. As this continues and as people learn more, they naturally start congregating with others who have similar loves. This way evil people eventually take themselves to hell, where they can be with others who share their evil. Good people, on the other hand, can be prepared for heaven.
Two important things have to happen for us to truly enter fully into heaven. First, the Lord will push aside our remaining evil desires, so they cannot hurt us or tempt us anymore; angels are in a marvelous state of peace, with no active evil to trouble them. Second, we will each be led by the Lord to the perfect married partner, the one whose soul matches ours, the one we can love blissfully to eternity. All angels are married, because the marriage of a man and a woman represents the marriage of love and wisdom in the Lord, and also the marriage of the desire for good and understanding of truth in each of us. Because of this, we can only fully receive and return the Lord's love as married partners, and heaven is suffused with the sphere of marriage and the love of marriage.
The angelic couples will find their way to communities of other angels whose loves match their own, people with whom they can share the deepest friendships imaginable. They will have houses which reflect the character of their loves, and will be given work to do that springs from their loves and fills them with joy. Beyond that, their lives are much as life might be in this world, though free of sickness and aging and boredom and conflict. They have bodies that are human in form - no wings! - but a beauty in face and form that reflects the good loves they have inside. They eat and drink and laugh and sleep and have parties and games; all filled with the delight of mutual love.
The Writings tell us the work angels do is varied far beyond what we can imagine, though they only describe a few aspects. Among other things, angels care for people in this life, passing on to them true ideas and desires for good from the Lord. They also teach those in the World of Spirits, greet those who have just died, raise those who died as children, keep order in hell and do many other things.
We would finally note that there are three degrees of angelic life, based on the loves people embraced in this life. The first, lowest heaven, called the "natural heaven," is filled by those who are in the love of service. Angels there love to do what's right because they know it is right. The second, middle heaven, called the spiritual heaven, is filled by those who are in the love of the neighbor. Angels there love to engage their minds with spiritual questions to gain an ever-deeper understanding of how to be loving to one another. The third, highest heaven, known as celestial, is filled with those who are in love of the Lord Himself. From that love they have such innocence that they look like children, and they instantly perceive what is true, in all its variety, from the light of that love.
(References: Apocalypse Revealed 818; Arcana Coelestia 228-233, 454, 1802, 2551, 2572 [3-4], 5470, 6872 [2-3], 8747, 9503 [1-3], 9814 [2], 10604 [2-4]; Conjugial Love 44 [6-10], 52; Divine Love and Wisdom 19, 63, 71, 115, 116, 202, Divine Love and Wisdom 321, 322, 334; Heaven and Hell 75, 133, 266, 267, 304, 311, 415)
