Moed Katan, Chapter 3, Mishnah 1. Before we turn to the Mishnah itself, we should preface that the Sages forbade cutting one's hair, shaving and laundering clothes on Chol HaMoed. The reason for the prohibition: so that a person will not put off his shaving, his haircut or his laundry until Chol HaMoed, when he is free from his work and has time on his hands - and as a result he would enter Yom Tov unkempt and not looking as he should.
Therefore a person may not do these things on Chol HaMoed; rather, he must take care of them in advance, before the festival begins. There are, however, exceptions, and these are the cases discussed in our Mishnah and in the Mishnayos that follow.
The text of the Mishnah:
"V'eilu megalchin bamo'ed" - these are the people who are permitted to shave and cut their hair on Chol HaMoed:
"Haba mimedinas hayam" - one who arrived from overseas on Chol HaMoed, or right before Yom Tov, and had no opportunity to get a haircut beforehand. This leniency applies specifically to someone who traveled overseas for business or for another necessary purpose; but one who traveled merely for pleasure has no adequate excuse.
"V'hayotzei mibeis hashivyah" - one who had been taken captive by non-Jews.
"V'hayotzei mibeis ha'asurim" - one who was released from prison, even a Jewish prison. In both of these cases he is permitted to cut his hair, since while he was held in captivity or in prison a person does not do so, because he is in distress and in a difficult situation. Once he goes free, he is permitted to shave even on Chol HaMoed.
"V'hamenudeh shehitiru lo chachamim" - one whom the beis din placed under a ban of excommunication, and who had the ban lifted on Chol HaMoed. A person under a ban may not take a haircut, cut his hair or launder his clothes, so he was unable to do so beforehand; once the ban is lifted, he is permitted to shave.
"Umi shenish'al l'chacham v'hutar" - one who took a vow not to cut his hair, and had no opportunity to go to a sage to annul his vow, or did not take advantage of the opportunity he had, or was unable to find grounds for releasing himself from the vow - until Chol HaMoed came and he approached a sage and the vow was annulled.
"V'hanazir" - a nazir whose days of nezirus ended on Chol HaMoed. The halachah is that shaving the head is part of the process of concluding nezirus, and therefore he is permitted to shave even on the festival.
"V'hametzora ha'oleh mitum'aso l'taharah" - a metzora moving from a state of tumah to a state of taharah, whom halachah requires to shave his hair as part of his purification process, and he is certainly permitted to do so on Chol HaMoed.
In summary: The Sages forbade haircutting, shaving and laundering on Chol HaMoed, so that a person would not put off his preparations and enter Yom Tov unkempt. This Mishnah lists six exceptions, all of which share the reason that the person had no opportunity to shave in advance, before the festival: one who arrived from overseas, one released from captivity, one released from prison, one whose ban of excommunication was lifted by the Sages, one whose vow was annulled by a sage, the nazir, and the metzora moving from tumah to taharah.