It's a loosely defined term, meaning High Density Interconnect. Usually it's for boards that have smaller than 0.3mm size vias, less than 0.15mm wide tracks and clearances, blind or buried vias, more than 8 layers. Any of these would place you in the HDI category, which makes the board more expensive.
If you are worried about your vias, just make them larger than 0.3mm hole, and 0.75mm pad, less than a few hundred vias, and most PCB houses will make the board without extra charge.