What does "no clean" solder actually mean?
Soldering requires flux to dissolve oxides and to promote wetting. Rosin (made from tree sap) is one flux material that has long been popular, and comes in various strengths- RMA (Rosin Mildly Activated) or RA (Rosin Activated). Kester says you can leave the flux on the board under fairly benign conditions and experience bears that out (it only becomes active = corrosive) at elevated temperatures, however most manufacturers will clean the board for cosmetic reasons and to allow inspection. Cleaning often involves the use of petroleum solvents- for example by vapor degreasing or just scrubbing.
Aqueous clean fluxes have been developed that can be cleaned without solvents- using just hot water and detergent.
No clean fluxes allegedly don't need to be cleaned and can be left on the board, however many of us have had problems with no-clean processes having relatively conductive residue. The residue is ironically extremely hard to remove, much more difficult than the two above-mentioned processes, more like 'can't clean'. Think twice about this kind of flux if you're thinking of sensitive analog boards that have high impedances. Even circuits you may not think of as being analog such as RTCC chips with an external crystal may be affected.
Safest for sensitive boards is rosin flux followed by a thorough cleaning process.
It doesn't apply so much to the solder, but to the flux incorporated into the solder. It means that it is not necessary to clean any of the stuff up that is left by it.
What is left is e.g. resin and for most applications it doesn't matter that it is still there.
So what is then the other kind of flux you may ask? It is much more aggressive, and may damage the material it is left on or the solder or components.
However even no clean flux often has enough "low" resistance that it might influence sensitive circuits, so in case you can live with a few hundreds of megohms resistance between anything, leave it on. If that is too low, clean it up.
The substance left behind is flux, which cleans the metal pads on the board when heated in order that the solder has a clean surface to bond to and you get the best possible quality joint.
The "no-clean" aspect means that leaving this in place will not damage the board. Other flux types can typically be very corrosive, and over time may etch through tracks leaving the board non-operational. No-clean is useful in prototyping as it removes the need for cleaning, although for aesthetic or "I don't want this sticky stuff on my board" purposes, you may still want to.
Normally no-clean flux is not water soluble (there is water soluble flux available, but it is not no-clean...). I usually use a dedicated flux cleaner (Fluxclene is one brand, but many are available). Alternatively, isopropyl alcohol works well.