Swedenborg generally uses this term in contrast to the terms "spiritual" and "natural." The celestial level (of heaven, of the inner meaning of the Bible, etc.) is that which is most concerned with the Divine and with love. It is the inmost or deepest degree.
Think of it this way: in some ways, the Lord is simplicity itself since, taken in His essence, He is love. His attitude toward us is also simplicity itself: He loves us. And He expresses that love in an unending stream that works completely and constantly to get us to freely accept and return His love. When you start looking at how that process works, it can start to seem complicated. But in essence, it could hardly be more simple.
It makes sense, then, that the best state we can achieve as humans is the celestial state in which we receive the Lord’s love and love Him back. It is the highest, best, purest, most innocent, and most joyful state human beings can experience, one powered by love of the Lord. It is also the state of the highest, inmost heaven, and is in some degree present in everyone as the recipient of the Lord’s love.
Not all of us receive and manifest the Lord’s love in the same way, though (in fact, none of us do), so the pure simplicity of the celestial state gets complicated as people open themselves up only partway to the Lord and redirect His love toward lesser things. Swedenborg’s works tell us that this results in three distinct “levels” of existence, based on what people love. The celestial is the highest of these levels, based on love of the Lord; the “spiritual” is based on love of other people, and the “natural” is based on the delight we feel in being good. The spiritual is more external and less pure then the celestial, and the natural is more external and less pure than the spiritual.
Each of those levels is further divided, however, between those who are led directly by the level’s defining love and those who are led by the ideas that spring from that love. Some celestial angels, then, are led by the love of the Lord itself, while others are led by the exquisite concepts that express love to the Lord. Unfortunately, Swedenborg also uses the words “celestial” and “spiritual” to identify that division, with “celestial” representing the love of each level and “spiritual” representing the truth and wisdom of each level. So the celestial level - centered on love of the Lord - has a celestial aspect and a spiritual aspect. The spiritual level - centered on love of other people - also has a celestial aspect and a spiritual aspect. And the natural level - centered on the delight of being good - has a celestial aspect and a spiritual aspect.
That might sound awfuly complicated - maybe even unnecessarily complicated - but it makes sense if you think about it. We all know people who are simply motivated by the desire to be good and to do things the “right” way. That’s an example of the natural level of existence, focused largely on external things, but aligned with the Lord’s wishes. Within that group, there are those who do what’s right pretty much by instinct, following their good affections; they would be the celestial natural. There are also those who like to know the rules and think about the instructions, so they can know intellectually that they are doing things right; they would be the spiritual natural. There are similar distinctions on the spiritual level and the celestial level.
One other thing is worth noting: Swedenborg’s works say that very few in the modern world are capable of reaching the celestial level, due to the amount of knowledge we have, the external nature of our lives and the need we have to use our minds to advance spiritually. The celestial heavens are largely populated by people from ancient times, who were able to live in closer, more direct communion with the Lord. As we understand the Lord’s wishes on a deeper and deeper level, though, we can open up greater possibilities for ourselves and for future generations.