First, let’s define relic:
236. The Second Vatican Council recalls that “the Saints have been traditionally honored in the Church, and their authentic relics and images held in veneration”( SC 111; Cf The Council Of Trent, Decretum de invocatione, veneratione et reliquiis Sanctorum, et sacris imaginibus (3 December 1563), in DS 1822). The term “relics of the Saints” principally signifies the bodies – or notable parts of the bodies – of the Saints who, as distinguished members of Christ’s mystical Body and as Temples of the Holy Spirit (cf. 1 Cor 3, 16; 6, 19; 2 Cor 6, 16)(Cf. SC 111) in virtue of their heroic sanctity, now dwell in Heaven, but who once lived on earth. Objects which belonged to the Saints, such as personal objects, clothes and manuscripts are also considered relics, as are objects which have touched their bodies or tombs such as oils, cloths, and images.
A relic is a body, or a “notable” body part. Not something that might be a piece of skin, fingernail, or mysterious little polygon. Likewise clothing and such: whole objects or significant and recognizable parts.
The Directory on Popular Piety and the Liturgy is online at the Vatican site.
Catholic Sensibility 
We used to speak of classes of relics, with bodily fragments being first class. I confess to a certain “ick” rection to first class relics; I don’t feel that way about 2nd or 3rd class. I know, it’s a cultural thing. But it does seem to contradict the Church’s teaching on the appropriate treatment of bodies of the dead, that they should be buried or entombed, and ashes should not be scattered or kept in one’s home.
Yes, I left room for people to comment on classes of relics. There remain three classes. Second class relics are notable objects associated with a saint. Third class relics are created just by touching an object to a saint’s tomb or other higher class relic.
And regarding the appropriate treatment of a dead body, that, I believe, was the issue of conflict between the dioceses of Peoria and New York over Bishop Sheen’s remains.
Careful; “principally signifies” is the important qualifier. It is not exclusive.