This is the most common method of collecting seeds.  It involves harvesting maturing seed heads and drying them for a period of time before crushing the seed heads and winnowing them to remove all but the actual seeds.  For a detailed explanation of this method see Saving Lettuce Seeds. 

Vegetables that you would use the dry method to collect seeds from include Lettuce, endive, carrot, Brassica plants (cabbage, broccoli Cauliflower etc..), leek, onions, beans and peas.

This involves washing the seeds in a sieve with a jet of water before tipping the seeds out onto newspaper to dry.  For a detailed explanation of this method see Saving Tomato Seeds. 

The main vegetables that you would use the wet method to collect seeds from are tomatoes, pumpkins, zucchinis and cucumbers.