In a purposive sample, you sample from a population with a particular purpose in mind. This is in contrast to a random sample, where you choose subjects in some random fashion, and also in contrast to a convenience sample, where you pick subjects based on some convenient factor (e.g., they happen to be in your class that day).
The main disadvantage of purposive sampling is that the vast array of inferential statistical procedures are then invalid. Inferential statistics lets you generalise from a particular sample to a larger population and make statements about how sure you are that you are right, or about how accurate you are. Although some methods have been developed for some purposive samples, they are more complex and not as well developed as those for random samples.
It is Easier to Get a Sample of Subjects with Partiuclar Characteristics
One way of doing a purposive sample is to find people who share particular characteristics. For example, if you had developed a new shampoo only for people with curly hair, you might want to find a sample of people with curly hair. It would be difficult, if not impossible, to get a full list of such people and take a random sample from them; if you sampled everyone and then asked everyone if they all had curly hair, you would waste a lot of time on people with other hair types.
Weighting for Unusual Characteristics
One type of purposive sample is a quota sample. In a quota sample, you look to get a particular number of subjects with particular characteristics. For example, you might be particularly interested in how Native American people voted in the last election but be still interested in how others voted. You could then sample so as to get at least 100 Native Americans. This would let you make more accurate statements about their voting behaviour and compare them to others as well.
Accessing People with Stigmatized Traits
If you need to get a sample of people who share some trait that is stigmatised (for example, using illicit drugs) then one method is snowball sampling. In this technique, each person in your sample recommends others who might be interested in taking part.