A couple of tips:
1. Use Yukon gold or red potatoes. They hold their shape best (they don't fall apart or crumble in the salad). Don't use Russet or baking potatoes.
2. Use plenty of water, and salt it. For a couple of quarts of water, add about 3 teaspoons of salt.
3. Peel the potatoes. Don't cut them too small. They should be in large chunks, bigger than bite size. The most important thing is that all the pieces are the same size, or else they'll cook unevenly, with the smaller pieces cooking before the larger pieces. (Peeling the potatoes rather than leaving the peels on allows the potatoes to absorb more flavor. I think it's easier to remove the peels before cooking, especially since you've cut the potatoes up, but that's personal preference.)
4. Put the potatoes in the water when the water is cold and bring the water to a boil. Turn down the heat and simmer the potatoes (gentle bubbles breaking the surface of the water, not fierce bubbles bursting and splashing, and you don't want the potatoes bouncing around like ping pong balls).
5. Check the potatoes after about 10 minutes, by piercing one gently with a regular fork. The fork should easily pierce the potato, but the potato shouldn't fall apart. If you're unsure, remove a piece, let it cool briefly, and taste it. It should be very tender, but still retaining it's shape. Continue cooking if necessary. Then drain the potatoes in a colander and let them cool just until you can comfortably work with them.
6. A basic classic potato salad recipe: Cut the cooled chunks of potatoes into bite-size pieces. Combine them with diced hard-boiled eggs, and diced peeled seeded cucumbers (cut the cucumbers the long way and run a spoon tip down the center of the cucumber so that the seeds scrape out - this keeps your potato salad from becoming watery or soggy). Add just enough real mayonnaise so that the potatoes are lightly covered but not swimming - the amount of mayo will depend on how many potatoes you used. It's always easy to add more mayo but impossible to remove it, so start slow and stir gently. Add about a tablespoon of Dijon mustard. Salt to your preferred taste. (Taste first before salting and add salt gradually).
7. Chill in the fridge and keep chilled while serving. You can fill a large serving bowl with ice and then set the potato salad bowl down into the ice so it stays chilled.
8. Don't put hot foods directly into the fridge. Allowing the potatoes to cool enough to handle comfortably, and then adding the mayo and other ingredients will usually lower the temperature enough so that you can put the potato salad into the fridge once all the ingredients are combined. Make sure to keep the salad cold and don't let it sit out too long at room temperature (no more than an hour or so, or 2 hours in the bowl of ice). It's not the mayo, by the way, that allows bacteria to grow - mayonnaise contains acid (vinegar or lemon juice) and commercially prepared (store-bought, brand-name) mayonnaise is safe at room temperature. However, when chicken or potatoes are combined with the mayonnaise, any bacteria on them (from our hands, or the knife we cut them with) lower the acidity level of the mayonnaise and after about 2 hours, the bacteria will grow. Or the mayonnaise gets diluted with the other ingredients, lowering its natural acidity. So it's not the mayonnaise - it's the combination of the potatoes or chicken or other vegetables or proteins mixed in with the mayonnaise that makes a potential bacteria circus. Keeping the salad properly chilled prevents this from happening. Homemade mayonnaise is a different story, and it's usually not safe at room temperature at all. But I'm talking about Hellmann's or Best Foods or Kraft, etc.