Watery potatoes

I imagine the problem is that the cells of the potatoes have absorbed too much water.

There are a few ways you can help avoid this:

• Boiling the potatoes with the skins on reduces the amount of water absorbed (and adds some of the flavour from the skins).
• Get the starch of the individual cells to gel before fully cooking. The method I have used involves keeping the potatoes at 70ºC (158ºF) for 30 minutes, cooling them and then boiling until cooked. There appear to be other methods.
• Use baked potatoes. This is the simplest and most reliable method if you don't have the equipment (or patience) for the previous method. Unfortunately it takes 2-3 times as long as boiling.

Update 1 (Overcooking): Although what I have written will help avoid water logged mash, the simplest answer is in the comment from SAJ14SAJ: the potatoes were overcooked. Here in Sweden it is common to use a thin metal skewer to test potatoes for doneness (see image). The skewer is about 6 inches long and 2mm thick. This makes it long enough to reach the center of pretty much any potato and thin enough not to crack or otherwise visibly damage the potato when testing (not that this matters for mash). In Swedish these are called potatisnålar (literarily potato-needle).

When the skewer can be pushed into the center of the potato without resistance, it is done.

Update 2 (Potato Variety): Some potato varieties have a higher water content, and the water content can differ within a variety. With that in mind, red potatoes are not necessary a good choice for boiling unless you try the starch gelling tip above.

You should look for potatoes with a medium to high percentage solids. The potato varieties available differ widely from country to country so I won't recommend a specific variety, but you can do a simple test to check the solid content of your potatoes using a saline solution with a 1:7.5 salt to water weight ratio and a clear container. If the potato floats about the middle of container, it is around average solid content. If it floats higher, it has a high water content. If it sinks lower, the solid content is above average.