A major topic... I'll focus just on vegetables, since they are most commonly saved.
When saving seed, the first issue is keeping the seed pure. If the variety is a hybrid, then the saved seed will not be true to what you grew... it might be fun to try, but not if you are counting on getting food, and it will be the only variety you grow. Open pollinated varieties will grow true-to-type if crossing is prevented; "heirlooms" are just open-pollinated varieties that have a documented history.
Even if you start with an open-pollinated variety, you will need to prevent crossing to keep the seed pure. Different vegetables vary greatly in their susceptibility to crossing; corn (wind pollinated) and squash (bee pollinated) are perhaps the most likely. There are different methods that can be used to prevent crossing, with varying degrees of difficulty... I'll mention several of those below. In many cases, seed will most likely be pure (or reasonably so) if only one variety per species is grown.
Seed should be saved only from healthy plants that have the traits you desire. Some seeds (such as beans, radishes, and okra) are harvested when the pods are dry; "wet" seeds (such as tomatoes, peppers, cucumbers, squash, etc.) should be harvested from fruits that are allowed to fully ripen on the plant. Wet seeds usually require some degree of processing, but this is fairly easy.
Once seed has been harvested & cleaned, it needs to be dried to a low moisture level prior to storage. No process is more critical to maintaining seed longevity than proper drying, and keeping moisture & humidity away from the stored seed. In my climate, where the indoor humidity is very low in Winter, just leaving the seeds exposed to the air in December will dry them enough. In warmer, more humid areas, it may be necessary to use a drying agent (desiccant) in a sealed container, to get seeds dry enough. As a rule of thumb, you can test seed dryness by hitting with a hammer (for beans and peas) or by bending the seed. If it shatters or breaks cleanly, it is probably dry enough for storage... if it just pulps or bends, it needs further drying.
Dry seed stored in paper envelopes at room temperature will have a seed life similar to that in the numerous charts which are online, such as the one linked by
@digitS' . Seed stored in air-tight containers will last considerably longer, especially if excess air is squeezed out. I recommend zippered freezer storage bags, since they use thicker plastic & tighter seals. My beans, stored in freezer bags at room temperature, still have good germination even after 7-8 years. Refrigerated seed would last even longer, and frozen seed can last for decades.
Proper dryness is especially important if the seed will be frozen, since excess moisture would allow ice crystals to form in the seed, which may destroy them. Frozen seed should be sealed in an air-tight container; and to prevent condensation from destroying the seed, it must be warmed up to room temperature before being exposed to air. For this reason, frozen seed is best stored in sealed plastic bags within a larger container, so that only the seed being planted needs to be thawed.
@Ben E Lou , I & others could provide more information, if we knew what seeds you are interested in saving. I can tell you from experience that squashes in the
C pepo species - which includes most summer squash, most pumpkins, acorn squash, spaghetti squash, and warted gourds - are very difficult to keep pure, since they are widely grown & bees will fly up to a mile or more for their pollen. Carrots can cross with wild carrot (Queen Annes Lace), and radishes can likewise cross with wild radishes. Corn is perhaps the most difficult to save seed from, even if starting with an open-pollinated variety (nearly all commercial varieties are hybrids). Isolation distances apply only if you intend to allow the plants to open pollinate; but there are many isolation techniques that can be used to reduce (or eliminate) the distance requirements. I'd be happy to provide some of those techniques for specific crops.
Oh, and I heartily recommend the book "Seed to Seed" by Suzanne Ashworth. It is a great introduction to seed saving, and to learning the species of most vegetables. Also, for great seed saving tips appropriate for Southern growers, I recommend the guides prepared by Jeff McCormick, formerly of Southern Exposure Seeds:
http://www.savingourseeds.org/publications.html
.
. You might be happy with your Jack o'lantern and think that it would be easy to save seeds. Your neighbor a block away has a zucchini plant. The next year, you are saying, "what the heck!?"
As others suggested, put those opened commercial seed packets into baggies, and then put those in a glass jar with a tight fitting lid. If you can add one of those moisture absorbing packets (from a prescription bottle) to that jar, even better. Put the closed jar in your freezer (frost free is best) and next year, when it is time to plant, take the whole jar out and let it warm slowly to room temperature and give any surface condensation a chance to dry before you open it. Your seeds should have great germination for a number of years, although some kinds (onion seed already was mentioned) simply don't last as long as others.
