I can't really explain precip. At least not the chemistry of it. Jsoong.... I know you can educate us all.
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I can try!
precip - short for precipitation - any process where previously dissolved material comes out of solution.
Eg. A solution of CaCO3 (calcium carbonate - what you get in hard water) can precipitate when water evaporates. Water can only hold so much of any material in solution. <You can only dissolve so much sugar or salt (or anything else) into water until no more can dissolve, the water is then said to be "saturated".> When water evaporates, only pure water is loss. You have less water remaining, but same amount of CaCO3, so some of it comes out of solution & become solid. You see that as white scales/deposits/particles in your tank/bathtub ....
Another eg. - Water vapour is "dissolved" in air. <Strictly speaking not true ... it is a gas mixture, not a solution.> With temperature change, etc. the water vapour turns into liquid (or solid) and "comes out of solution", falling as rain or snow. So rain, snow, sleet, etc is often called precipitation.
These are the simplest forms of precipitation. Minerals & salts can also come out of solution with pH changes, temperature change, or chemical interactions between ions. These processes are more important in SW tanks. Although simple evaporation is prob just as big a problem.
Back to the original question - I think that a bacterial bloom is the likeliest explanation. Until you get the proper bio-filter established, the excess ammonia causes other bacteria to grow, which floats around in the water. Clumps of the bacteria you see as white particles in the water.
Whether it is a bacterial bloom, hard water deposits (or precips) or a bio-film, all are perfectly harmless during your cycle. ATM, you should just concentrate on establishing the cycle. No need to worry too much about water clarity, etc. just yet.