The difference is in the kind of ink particle...the dye or pigment. In your case, PGI-* are pigment inks, and CLI-* are dye (ChromaLife 100 year) inks.

Dye inks use dye particles, which are soluble substances. Dye particles tend to be smaller than pigments, lay flatter on the surface of papers, and are often a bit more vibrant in the way they reflect light (which can actually be a detriment, rather than a bonus, when trying to print color-accurate photos.) Dye inks tend to do quite well on glossy papers and do not usually exhibit gloss differential. Dyes, as they are soluble, tend to have shorter lifetimes.

Pigment inks use pigment particles, which are insoluble substances simply suspended in solution. Pigment particles are larger, and can often lay unevenly on paper. Pigment inks can also exhibit certain effects such as metamerism, bronzing and gloss differential, however these days pigment inks are highly advanced, and such problems are mitigated to their minimums. Pigments also tend to be more controllable than dyes, and an extensive range of pigment inks is available for high quality pigment-based ink jet printers. Gamut and overall color quality that can be achieved by full pigment ink printers is second to none these days. Pigments, being non-soluble (and given their significant technological advances over the recent years), tend to have significantly longer lifetimes than dye inks.

The particular printer you have is not really known as a high quality photographic printer in the grand scheme of things. It probably has a pigment black ink to improve the permanence of text printouts, and possibly to support matte papers better (which have a tendency to absorb dye inks a bit too much, washing out the color.) The dye is probably used to support gloss better, as pigment ink usually causes a gloss differential...if pigment black is used on gloss with the dye colors, it usually leaves quite noticeable gloss differential for all the black (text, lines, etc.)