A divorced Catholic who is not remarried can usually receive holy Communion. Divorce is not intrinsically evil, and so it is not necessarily a sin. But divorce can sometimes be a serious sin, depending on the intention and the circumstances. A divorced Catholic should consult with his pastor or his confessor about whether or not he has sinned by getting a divorce.
Divorced and remarried Catholics are not allowed to receive Communion because they are having sexual relations with a person to whom they are not married in the eyes of God and the Church. If the prior marriage was the true Sacrament of Marriage, then the divorce cannot break the bond of the Sacrament. The couple remains married to each other in the eyes of God and the Church. So when that person remarries, he or she is having sexual relations outside of a valid marriage.
Divorced and remarried Catholics can sometimes obtain an annulment, if the previous marriage was, for some reason, not the true Sacrament. Another option is for them to cease from sexual relations, but stay together for the sake of the children. A third option is for them to separate. After any of these three options, with confession and permission from the Church, they may resume receiving holy Communion.