In the natural world context, blue ice is simply water ice. If ice is formed without air bubbles or debris contained inside it, it is mostly transparent to light; however, it absorbs light at the red end of the spectrum much more efficiently than light at the blue end. Hence, if you look at ice that is thick enough, it will appear to be blue, as the light that has penetrated partway before being reflected back out will have had more of the reds removed than blues - and appear blue. Glaciers, being one of the most likely places to find ice both thick and consistent enough, often appear blue beneath their surface white coating of snow or broken ice. For the best example, look into a crevasse.
momomom notes that there is another use for the term: flexible reusable freezer packs, which are typically filled with a blue substance. I'm not sure whether those are blue for safety, for marketing, or simple chance. Thanks momomom!
Of course, that's not all.
In the modern world, we have this thing called air travel. Air travel implies people on airplanes, often for hours. This, logically, requires the inclusion of toilet facilities aboard aircraft. Since it is highly inefficient to carry lots and lots of water aboard an airplane, airplanes use chemical toilets to maintain (mostly) hygienic interiors. This requires less space. It is possible, however, for some aircraft to dump the contents of their waste systems in order to reduce weight - and make more space inside said systems if necessary.
Well, the predominantly used chemical mixes in chemical toilets are deep blue. This is the result of a dye added to quickly identify the substance as sewage and therefore a contaminant, as well as identify any surface which has been contaminated with it.
When said tanks are flushed at the high altitudes airliners typically travel at, the contents will almost immediately freeze solid in the thin cold air.
However, if the dump hasn't occurred properly (and sometimes if it has, depending on the design) then rather than a fine spray of chemical yummy, the aircraft will form chunks of frozen chemical waste (containing guess what!) on the outside of the airplane near the release valve. Eventually, through vibration, becoming too large, or moving into warmer climes, those chunks will break away.
Every year, there are a few news stories about unlucky groundlings who are visited abruptly by "blue ice" - news industry euphemism for meteoric frozen chemical-frosted turds.