Plainsong, plainchant and Gregorian chant in common usage mean the same thing: the kind of music used by monks and nuns in the liturgy. Chant has only a single melodic line and is usually sung unaccompanied. Its rhythm is very free as the music is shaped and formed according to the accentual understanding of the text being sung.
The origins of the chant we use today probably go back to the synagogue music of the first century. It is thus a sacred bridge between the early church and our own day.
Different traditions of chant have survived but the most important has proved to be the Roman. In the nineteenth century the monks of Solesmes began a scholarly revision of the most-used liturgical books which continues to this day.
To the non-specialist perhaps the most striking thing about the chant is that it is written on a stave of four lines. It is, however, music that is wonderfully expressive and the revival of chant is to be encouraged.
The Musica Sacra site of the Church Music Association of America is an excellent starting-point for discovering more about the chant. There are audio files you can listen to, as well as downloads of music and texts for liturgical use.
Michel Ozorak has produced a series of free online chant sheets intended to help clergy with the singing of the Mass prayers and readings. You can access them at the Assumption of Our Lady, Windsor, web site here.
If you are lucky enough to own an iPod Touch or an iPhone, there is a very useful app which allows transpositions on the fly (no more fumbling with pitch-pipes), iGregorian.
A brief historical bibliography:W. Apel, Gregorian Chant (1958);
D. Conomos, Byzantine Hymnology and Byzantine Chant (1984);
D. G. Murray, Gregorian Chant According to the Manuscripts (1963);
R. / B. C. Pugsley, The Sound Eternal (1987);
J. Rayburn, Gregorian Chant (1964);
Solesmes, ed., Paleographie musicale (1889), MS Einsiedeln 121 (1894), and vol. 10, MS Laon 239 (1909);
S. J. P. van Dijk, "The Old - Roman Rite," Studia patristica 80 (1962), "Papal Schola versus Charlemagne," in Organicae Voces (1963), and "The Urban and Papal Rites in Seventh and Eighth Century Rome," Sacris erudiri 12 (1961);
J. W. A. Vollaerts, Rhythmic Proportions in Early Medieval Ecclesiastical Chant (1960);
P. Wagner, Introduction to the Gregorian Melodies: A Handbook of Plainsong (1910);
E. Werner, The Sacred Bridge (1959).