Communion under both kinds refers to receiving both the bread and wine at communion, specifically by the laity.
Biblically, those partaking in communion should receive both the bread and wine, as the Lord explicitly said "all" present were supposed to drink the wine (Luke 22:17):
Matthew 26:27
27 And he took the cup, and gave thanks, and gave it to them, saying, Drink ye all of it;
Similarly, the following passage indicates that the average person was partaking of the cup during communion:
1 Corinthians 11:25-28
25 After the same manner also he took the cup, when he had supped, saying, This cup is the new testament in my blood: this do ye, as oft as ye drink it, in remembrance of me.
26 For as often as ye eat this bread, and drink this cup, ye do shew the Lord's death till he come.
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.
Historically, also, during the 1st-12th centuries, communion under both kinds was practiced:
Catholic Encyclopedia - Communion Under Both Kinds
It may be stated as a general fact, that down to the twelfth century, in the West as well as in the East, public Communion in the churches was ordinarily administered and received under both kinds.
Prohibiting communion under both kinds, in light of the above, would be contrary to both the Bible, and the near-unanimous tradition of the Church for the first 1,000 years.
Communion under both kinds was explicitly forbidden, on pain of excommunication, by the ecumenical Council of Constance (1414-1418 AD):
Council of Constance - Session 3 (1415 AD)
Certain people, in some parts of the world, have rashly dared to assert that the christian people ought to receive the holy sacrament of the eucharist under the forms of both bread and wine. They communicate the laity everywhere not only under the form of bread but also under that of wine, and they stubbornly assert that they should communicate even after a meal, or else without the need of a fast, contrary to the church’s custom which has been laudably and sensibly approved, from the church’s head downwards, but which they damnably try to repudiate as sacrilegious.
Moreover, just as this custom was sensibly introduced in order to avoid various dangers and scandals, so with similar or even greater reason was it possible to introduce and sensibly observe the custom that, although this sacrament was received by the faithful under both kinds in the early church, nevertheless later it was received under both kinds only by those confecting it, and by the laity only under the form of bread.
[That no priest, under pain of excommunication, may communicate the people under the forms of both bread and wine]
This holy synod also decrees and declares, regarding this matter, that instructions are to be sent to the most reverend fathers and lords in Christ, patriarchs, primates, archbishops, bishops, and their vicars in spirituals, wherever they may be, in which they are to be commissioned and ordered on the authority of this sacred council and under pain of excommunication, to punish effectively those who err against this decree. They may receive back into the church’s fold those who have gone astray by communicating the people under the forms of both bread and wine, and have taught this, provided they repent and after a salutary penance, in accordance with the measure of their fault, has been enjoined upon them. They are to repress as heretics, however, by means of the church’s censures and even if necessary by calling in the help of the secular arm, those of them whose hearts have become hardened and who are unwilling to return to penance.
According to the above, a priest was strictly forbidden from giving communion under both kinds to the laity. Those who do so are excommunicated, and may even be remanded to the secular authorities.
This was affirmed by the Council of Trent (1545-1563 AD):
Council of Trent - Session 21 (1562 AD)
Canon 1 - If any one shall say, that, by the precept of God, or, by necessity of salvation, all and each of the faithful of Christ ought to receive both species of the most holy sacrament of the Eucharist; let him be anathema.
Canon 2 - If any one shall say, that the holy Catholic Church was not induced by just causes and reasons to communicate, under the species of bread only, laymen, and also clerks when not consecrating; let him be anathema.
The justification given for this doctrine is as follows:
In response:
As part of the liturgical reforms of the Second Vatican Council (1962-1965 AD), communion under both kinds was reinstated:
Vatican 2 - Sacrosanctum Concilium (1963 AD)
55 The dogmatic principles which were laid down by the Council of Trent remaining intact, communion under both kinds may be granted when the bishops think fit, not only to clerics and religious, but also to the laity, in cases to be determined by the Apostolic See, as, for instance, to the newly ordained in the Mass of their sacred ordination, to the newly professed in the Mass of their religious profession, and to the newly baptized in the Mass which follows their baptism.
The above made a formerly excommunicable offense - giving wine to the laity - permissible. Though, notice it is still seemingly to be limited to special occasions, and not made a standard part of each Mass. However, the fact that it is permissible at all demolishes one of the justifications discussed earlier for having banned it - that distribution of wine to the laity has practical problems. Allowing it has demonstrated that the Church knows that accommodations can be made to overcome these problems.
Defenders of Catholicism will justify the above contradiction on the basis that it is a matter of discipline, not faith and morals, which is an excuse that gets quite tiresome for those of us who deal with these matters often. In this case, it is especially ridiculous:
What changed in 1963 that made the laity able to partake in communion under both kinds? Why was following this Biblical and historical practice something that would get one thrown out of the Church from 1415-1962, but not in 1963?
Banning communion under both kinds makes absolutely no sense, Biblically or historically. Someone concerned with integrity before God must simply acknowledge that the Council of Constance erred.
However, once one has declared themselves to be infallible based on errant theological conclusions, suddenly everything that one has ever declared must be bitterly defended, no matter how blatantly indefensible it is, or even if it was later contradicted by another "infallible" council.