Positivism is usually (but not exclusively) associated with quantitative methods. Positivism has origins in the approaches of Auguste Comte (1798-1857) who coined the terms "positive philosophy" and "sociology", and Emile Durkheim (1858-1917) who advocated positivist methods to sociological enquiry. So social facts are like objects in the natural world, there to be studied.
Observation of social facts lead on by this inductive method to statistics. Using statistics leads on to correlations or causal associations between one observable fact and another. When a correlation is found between two sets of statistics then one suspects causality from one to another.
To overcome a spurious correlation Durkheim suggested multivariate analysis where one variable is tested against another. Is one independent variable more or less important on another independent variable on a set of dependent variables? So one hold's one variable constant and changes another, and see what happens to dependent variables.
Another important element of this method is repeated checking. Then we produce laws of human society.
The criticism of this empiricist approach is that it removes the need for interpretive meaning. Because there are external laws that govern social behaviour there is no requirement for people to interpret.
Karl Popper's approach is slightly different. It is a deductive approach where you start with a hypothesis, any hypothesis. Unlike with the hard line positivists, where the hypothesis must come from observable data, it doesn't matter where the hypothesis comes from. However, the hypothesis should be precise and testable. This hypothesis may stand the test of time or at some point fail - or become falsified in time. Scientists almost have to seek out evidence to disprove their hypotheses, in order to make them more rigorous.
A related method is comparative. In this a social researcher, unable to go into a laboratory or alter quite as many variables as a scientist might, compares between situations. The more positivist will attempt to be systematic in comparisons and make social laws. One isoloates the differences between groups so as to show what different variables are in operation and what different outcomes are shown. All founding sociologists were comparativists, including Durkheim himself.
However, comparison is not limited simply to positivist approaches. It can form part of an interpretive approach, where qualitiative data is preferred to quantitative.
For interpretive researchers, people are very different from objects of scientific research because people are self conscious beings. Quantitative data can never reveal what matters - the conscious subject's interpretations and perceptions. Meanings are constructed, they are not objects in themelves. Causal states suggested by positivists can only be the case when we know something of the meanings attached by individuals and cultures themsevles, and meanings can change.
Now some see the study of meaning (Weber, 1864-1920) as part of getting back to studying causal relationships on a large scale. Others see meaning as a subject in itself. For these latter people, you cannot isolate variables. What matters is the actor's view of the situation they are in, and all the researcher can do is absorb him or herself into that situation. They have to get right into a data rich environment and develop empathy and understanding top a point where they can give an account of the meanings of the situation. Such people are symbolic interactionists. However, a symbolic interactionist still thinks there is a social order, its just that the method has to be highly local and research rich.
Phenomenologists, however, do not believe there is a social order to be discovered. They believe that there is only meanings in the subjects. Statistics are themselves just statements of meaning - the meanings that are assumed are important in the making of statistics. Who is it that makes statistics and why? Who is it that acts and why? There are real problems with this method of going from the particular to making any general points. The most interpretative and phenomenological social anthropology can only absorb the researcher and produce description - even comparison is fruitless. Indeed it may be impossible to describe one culture in the terms of another inhabited by academics not involved in the field research.
Most research is a mixture of these methods and insights. Which approach seems to bear most weight usually depends upon the scope and question involved. Usually you have to have some focus but to have a hypothesis does not imply a deductive Karl Popper method, but rather this is a starting point and way to organise research activity. Falsification never quite wipes out a theory first time; it is usually tested further, and given the fluidity of social actors this must be the case in Sociology. After a hypothesis is set, anything can happen.
Haralambos, M,. Holborn, M., Sociology: Themes and Perspectives, London: Collins Educational, 698-711.