As a practical answer from a perspective of aquarium lighting, HQI usually means a double-ended lamp, versus a single-ended (screw in) lamp.
There are of course exceptions to this, but again, from a practical perspective, that's the way to look at it.
HQI lamps produce more light output for a given wattage than standard halides. They lamps REQUIRE an HQI ballast, but HQI ballasts can also run SE lamps for a nearly identical performance. Ergo, HQI is more a ballast spec than a lamp spec.
HQI lamps are used almost exclusively in commercially manufacturered strip light systems. They are also used in pendants. SE lamps are used in some pendants as well, and they are used almost exclusively by custom made (DIY'd) systems, in part because they require external shielding to be operated safely.