Very basically, quantative research is based upon collecting a large volume of data e.g. to identify a trend, qualitative research is based more on in-depth questioning but with fewer respondents / interviewees.
In theory quantative is logical & objective, qualitative is instinctive & subjective.
In practice, it's more complicated than that & research can often be a combination of the two.
Qualitive research is more suitible when dealing with issues that relate to understanding opinions or attitudes. Quantitative data is more suitible for issues that are mneasures statistically.