Over the course of the series, there has been a lot of ambiguity about how the avatar actually works. Many times it seemed as though each avatar was a seperate person, (they kept talking to each other) and others like they were all the same (they kept refering to reincarnation and past lives), and not to mention the avatar state. We'll never know for sure until those alleged and long-awaited episodes about the avatar lineage come out in book 2 spirits, but here are some basic theories I've run across.
-Each avatar is a seperate person. They really don't have a connection, the current one just finds the old one in the spirit world and gets advice.
-The avatar is a spirit that passes to the next incarnation. Basically, in the body of the avatar, there are two souls: that of the avatar spirit, and that of the person carrying it and serving the world (korra, aang, roku, kyoshi, ect)
-The avatar is one spirit, each is the same person.
I've come up with my own theory that, as far as I can tell, explains all of the ambiguities in the series. Check it out and let me (and the rest of us too!) know what you think - if you see any holes in my theory or want me to explain a particular situation, let me know so i can try to fix/explain it.
-The word 'reincarnation' means that the soul passes to another body to live another life. This term has been used so many times that we cannot ignore it. The avatar is a single entity. What about all the times Aang talked to Roku? When the avatar reincarnates, his soul passes on to a new life, and the memories from his old life become part of his unconscous mind, which resides in the spirit world, or appear there when the current avatar goes there. My proof of this is Roku's statement "I am a part of you." Roku is a part of Aang's (well, now Korra's, but i'm mostly going to refer to aang, because korra didn't have many spiritual expierences) unconcscous mind. (sorry about my spelling). Also, Roku told Aang, "Aang, you are me." Again, proof that they are two parts of the same mind. Think of the movie Inception. When in a dream, your unconscious (in the movie they said subconscious, they were wrong) is projected as other people (they were cleverly called projections) in the dream. This is how I imagine the spirit world to be like - when the avatar goes there (or calls their spirits from his body), his past lives are projected for him to talk to.
Before he has learned to control the avatar state, his unconscous mind is controlling his conscious while in the avatar state (like all the times aang activated it our of danger or emotion). The process of learning to control the avatar state is the process of learning to control your unconscious mind, which is what happens when the avatar uses the avatar state at will. The reason for the 'power boost' from the avatar state is that the avatar state is full access to his full mind and all his knowledge and power, unconscious or not. Normal people cannot do this. (real world too- (sort of) - study psychology).
When the avatar's body takes on the form of a particular past life (like when Aang chanelled Kyoshi in "Avatar Day") his mind is simply switching which lifetime's set of memories are conscious. When he calls a past life out of his body (as in the sozin's comet finale) he is projecting unconscous memories for him to talk to (much like what happens in the spirit world). When Korra was kidnapped by tarlock and had memories of Aang, what was basically happening was that Korra's memories from that lifetime were coming to her conscous mind without her having to project them. She is the first incarnation of the avatar that we have seen able to do this.
I just though of this part and didn't know where to put it so here it is: 'avatar' is derived from the Sanskrit (again, sorry about spelling) word "avatara" which, according to this wiki, means a descent of a god to earth in mortal form. This points to the belief that the avatar existed in the spirit world prior to coming to the physical world, and the fact that they are all the same person. (and as a side note, by that definition, the koi fish could be considered avatars of the moon and ocean spirits).
I hope this clears up some doubt for some people, or at least gets them thinking. Remember though, this is just my own theory and speculation based on evidence. Enjoy, thanks for reading, and don't forget to comment!
oh yeah, p.s. - look at the talk page of the avatar article. This issue is discussed there as well.