Growing plants from seed is a useful survival skill because it reduces the need to rely on outside sources for a food supply. Eventually any amount of stored food will be used and growing your own food can reduce the amount of stored food that needs to be eaten in a survival situation. By saving seeds from plants that are successful in your local growing conditions the plants will become better adapted to growing there.
The specifics of harvesting seed and storing it in a manner that will allow for germination in the next or subsequent growing season are different for each climate and plant.
Basic Seed Saving Considerations
While seed saving does not need to be complicated or high tech, there are a few things to consider when choosing what seeds to save and which methods to use.
Generally speaking, you will wish to save the seeds of open pollinated fruits and vegetables. Open pollinated plants have been around for many (plant) generations and are considered "stable" - in other words, their seeds will almost always grow up to be nearly exactly like their parent plants. All heirloom seeds are open pollinated; however, there are also many non-heirloom seeds that fall into this category. When sourcing your original set of seeds, you will wish to acquire them from a knowledgeable and reliable place. Seed catalogs will generally indicate which seeds are open pollinated.
Without going into the genetics of it, in addition to using open pollinated plants, you will need to know if your plants are prone to genetic depression - meaning that the plant quality will likely erode unless you save seeds from many, many plants. For example, beans and tomatoes do not exhibit this behavior, so you might be able to save seeds from only a few plants each year without any issues. With corn, however, it is recommended that one saves seeds from hundreds of plants to insure enough genetic diversity. As you can probably guess, seeds from beans and tomatoes will require significantly fewer plants than something like corn.
Then there is the issue of the actual pollination. You could plant two types of open pollinated pumpkins next to each other, and while the pumpkins that are produced will look and taste like their parents, the next generation of the pumpkins from those seeds will be unlike either one. Therefore, some method to ensure that your plant is only pollinated by the same type of plant is necessary. The three main methods are to separate the blooming of different plants by time, distance, or physical barrier. The other is to choose plants whose flowers will generally self-pollinate prior to the flower actually opening. Using self-pollinating plants is not a fool-proof method, but I have found that it works well with both beans and tomatoes.