Holy Communion is a common title often given to what the Bible specifically calls "the Lord's supper" (1 Corinthians 11:20). The phrase, Holy Communion, is not in itself a faulty designation. In 1 Corinthians 10:16 we read, "The cup of blessing which we bless, is it not the communion of the blood of Christ? The bread which we break, is it not the communion of the body of Christ?" Paul is definitely referring to the Lord's Supper as "communion" and it is a proper conclusion to think of it as "holy."
However, I do use this designation. The reason has to do with association. Many, probably most, churches that use the phrase Holy Communion think of the Lord's Supper as what is called a sacrament. The Bible does not called the Lord's Supper a sacrament. Rather, it is called an ordinance (1 Corinthians 11:2). The distinction is critical. An ordinance is a practice given to the churches which is to be followed according to God's revealed "order" (a word related to ordinance). A sacrament, however, is a practice or ceremony thought to supply a certain amount of grace or holiness to the one who keeps it. Historically, it has been a part of works salvation for Roman Catholicism and for Protestant churches that have not been able to cast off this teaching. More than that, it is unbiblical.
Why do we partake of the Lord's Supper? First of all, we do it because Jesus commanded His disciples and followers to do so. The partaking of the Lord's Supper was instituted at His Last Supper.
Matthew 26:26 And as they were eating, Jesus took bread, and blessed it, and brake it, and gave it to the disciples, and said, Take, eat; this is my body. 27 And he took the cup, and gave thanks, and gave it to them, saying, Drink ye all of it; 28 For this is my blood of the new testament, which is shed for many for the remission of sins.
Paul looked at this event as establishing the ordinance of the Lord's Supper. 1 Corinthians 11:23-24 states, "For I have received of the Lord that which also I delivered unto you, That the Lord Jesus the same night in which he was betrayed took bread: And when he had given thanks, he brake it, and said, Take, eat: this is my body, which is broken for you: this do in remembrance of me." Paul received the ordinance from the Lord Jesus and delivered it to us.
Second, we take of the Lord's Supper because it commemorates the death of Jesus Christ for our sins. Jesus said that we are to continue the practice "in remembrance of me" (Luke 22:19; 1 Corinthians 11:24, 25). The bread reminds us of the body of Christ that was broken for our sins and the juice of the grape reminds us of the blood of Christ that was shed as a payment for our sins. This is certainly a holy remembrance.
Third, we partake of the Lord's Supper in anticipation of His future coming. In Matthew 26:29, Jesus said concerning the Lord's Supper, "I will not drink henceforth of this fruit of the vine, until that day when I drink it new with you in my Father's kingdom." Notice how He looks forward to the day when He will drink with us in the kingdom. In 1 Corinthians 11:26, Paul tells us, "For as often as ye eat this bread, and drink this cup, ye do shew the Lord's death till he come." We do it while looking forward to the coming of Christ for His own. So, there are three reasons for the Lord's Supper: 1) obedience to the practice established by our Lord Jesus; 2) a backward look to the supreme sacrifice made by Him; and, 3) a forward look to His return when He will take the saved to be forever with Him.
You also ask who should not take of the Lord's Supper. First of all, it is not for the unsaved. It is for the purpose of commemorating what Christ has done for us. If we are not redeemed by the blood, the Supper has no meaning. Second, it should only be taken by those who partake for the right reason and in the right way.
1 Corinthians 11:27 Wherefore whosoever shall eat this bread, and drink this cup of the Lord, unworthily, shall be guilty of the body and blood of the Lord. 28 But let a man examine himself, and so let him eat of that bread, and drink of that cup. 29 For he that eateth and drinketh unworthily, eateth and drinketh damnation to himself, not discerning the Lord's body. 30 For this cause many are weak and sickly among you, and many sleep. 31 For if we would judge ourselves, we should not be judged.
Here, we are warned against partaking of the Lord's Supper "unworthily." Many have misunderstood this passage and have refused to take of the Lord's Supper because they feel that they are unworthy to do so. The fact is, we are all unworthy of the sacrifice made by Jesus Christ and are therefore unworthy participants in the Lord's Supper. But this is not what this passage teaches. The word, unworthily, is an adverb. That means that it describes a verb or further defines an action. This passage is not saying that we must be worthy of the Lord's Supper. It is saying that we must partake of it in a manner worthy of this special ordinance.
This fact is further explained in the context of the passage. Listen to Paul's complaint in 1 Corinthians 11:21-22: "For in eating every one taketh before other his own supper: and one is hungry, and another is drunken. What? have ye not houses to eat and to drink in? or despise ye the church of God, and shame them that have not? What shall I say to you? shall I praise you in this? I praise you not." The Corinthians had made a party out of the Lord's Supper and some people were hoarding the food they had brought while others were getting drunk. This angered the Lord so much that He had brought come of them to sickness and others to an early grave. They had made a mockery of His Supper.
So, when we come to partake of the Lord's Supper, we need to examine our hearts ("But let a man examine himself" - 1 Corinthians 11:28) to make sure that we are coming for the right reason. We need to make sure that we are "discerning the Lord's body" (1 Corinthians 11:29). To discern means to separate mentally from other things. That is, we should treat the taking of the Lord' Supper as a special event which commemorates the unique sacrifice that Jesus Christ made for us on the cross. However, if we seriously approach the Supper and seriously consider what Christ did for us on the cross, we are then partaking of the Lord's Supper worthily.
Thirdly, the Lord's Supper is specifically for the members of a local assembly. It involves a communion with the Lord Jesus and with each other. I will not get into the controversy of open, close, or closed communion here. I am only saying that it is a local church ordinance and is not meant for any and every meeting of believers. Paul says that the Lord's Supper is kept "when ye come together in the church" (1 Corinthians 11:18). Though some may argue the level of exclusiveness, it is still clear that it is a church ordinance. It is not an ordinance to be practiced outside the local church.