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Qualitative research is primarily exploratory. It is used to gain an understanding of underlying reasons, opinions, and motivations. Qualitative research is also used to uncover trends in thought and opinions and to dive deeper into a problem by studying an individual or a group.
Qualitative methods usually use unstructured or semi-structured techniques. The sample size is typically smaller than in quantitative research.
Example: interviews; focus groups
More info on qualitative research in the social sciences
Quantitative research is characterized by the gathering of data with the aim of testing a hypothesis. The data generated are numerical, or, if not numerical, can be transformed into usable statistics.
Quantitative data collection methods are more structured than qualitative data collection methods and sample sizes are usually larger.
Example: surveys; statistical analysis
Note: The above descriptions of qualitative and quantitative research are mainly for research in the Social Sciences, rather than for Natural Sciences as most natural sciences rely on quantitative methods for their experiments.
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