Here are a few of the basic disagreements regarding communion: symbol vs. sacrament, necessary elements, frequency, who can partake, and the ultimate purpose of it.
Again, I will show my hand at the start. I see communion as a sacrament (a rite that conveys grace, blessing, or holiness to the believer who participates in it). The elements are bread and wine (I am okay with grape juice as long as we understand that unfermented grape juice didn't exist until Welch's invented it). I am a proponent of it happening every week (even though my denomination usually only does it once a month). I think it should only be for those believers who have joined mystically with the Body of Christ (the Church universal) through baptism. I think the purposes of communion are:
1) Renew and strengthen the Church
2) Offer a sacrifice to God
3) Mystically connect with every other believer
4) Mystically connect with Christ
5) Point to the day when we will eat with God at the banquet table
I will now reveal more of my hand. Despite my seemingly strong and somewhat nit-picky beliefs, I also think that we must avoid the temptation to make it too formulaic. While I think there are proper ways of participating in communion, the goal is to participate. If you don't have wine or grape juice, use whatever you have. The point is to eat together. The point is to open up a place for Christ to be present. The point is to be strengthened. The point is to experience some of heaven on earth.
Christ said, "This is my body." Now, I am not much of a literalist, but I do believe that Jesus meant that in some way, he is Truly present in communion. Not just in the way that he is always there, but that in the act of the words being said, the elements being offered, and the people being gathered together Jesus is actually present. This is known as the real presence. Did you know that not only do Catholics and Orthodox believe this (Catholics define it as transubstantiation which means that the elements become the literal flesh and blood... the Orthodox do not have this defined the way Catholics do, but they are similar), but both Luther AND Calvin taught forms of the real presence? Luther taught consubstantiation. This means that Christ is present with the elements. Once the meal is done, he is not longer present. This is why he had a problem with the Catholic practice of eucharistic adoration. I don't have a problem with that practice since they are approaching it from the standpoint of believing that they are worshipping Jesus. If they believed he wasn't in the bread and still worshipped it, that would be idolatry, but it is not the way Catholics do it! Calvin said that Jesus was present in the meal by the preaching of the word and the blessing of the offering.
All three positions teach that grace is received through communion. Not the "saving grace" we think of when we normally hear the word grace (necessarily), but more the grace of experiencing God more and being further equipped because of the experience to love the world more. The position that it is merely symbolic loses this. It reduces the table to just a mere act of remembrance. While that element is definitely there ("do this in remembrance of me"), it is only the beginning. Again, as with baptism, stopping at the symbols is failing to see the bigger picture.
Because grace is conferred and the meal itself is unifying, there is absolutely no good reason not to do it every week (hell, the Catholics have it even better by doing it everyday)! Do we not want too much grace? Is too much unity unhealthy?