Cracked Pinblock/Loose Tuning Pins
The pinblock is a laminated piece of wood several inches thick that is secured behind the plate. The finely threaded tuning pins are pounded through holes in the plate into holes in the pinblock. There are two common reasons why tuning pins become loose.
"Worn out" pinblock
Most commonly, the pins have become loose from decades of having them turned back and forth in the holes during tuning, much like a wood screw that is stripped.
In this case, the pins can often be pounded a little further into the pinblock, which will give the piano many more years of life before anything more needs to be done. In grand pianos, before pounding the pins in, the action must be removed and a special pinblock jack must be set up to support the pinblock to prevent the laminations in the pinblock from splitting.
If the pins are already in as far as they should go, the piano can be restrung with over-sized pins and the piano will usually have another 40 or so years of life. An old treatment that was popular at one time was to apply a glycerin solution to the pinblock which would swell the wood up and sometimes get the pins to hold for a while. However, this makes the wood mushy and might make it impossible to use over-sized pins in the future. More recently, technicians have experimented with applying various types of glues to the pinblock with good results, however I do not have experience with this technique. Only in the case of a piano that is completely untunable and is going to be either thrown out or completely rebuilt at a later date would I risk "doping" the pinblock.
A more serious problem is when there is a crack in the pinblock between two or more tuning pins. The crack causes the holes to open up and the pins to become so loose they will not stay in place when tension is placed on them. Cracks in the pinblock can be caused by a poor quality pinblock, excessively dry environment causing the wood to dry out, and - in a grand piano - by an inexperienced technician pounding the pins in without supporting the pinblock from underneath with a jack, causing the laminations to split apart.
If a vertical piano has a cracked pinblock, it is often not worth the cost of fixing it. In a grand, the usual solution is to replace the pinblock, although this is quite expensive and would only be considered in a quality piano.
Some technicians have had success with removing all the strings and tuning pins, filling the holes with epoxy, and then drilling them out again.