Quantitative research refers to any research based on something that can be accurately and precisely measured. For example, it is possible to discover exactly how many times per second a hummingbird's wings beat and measure the corresponding effects on its physiology (heart rate, temperature, etc.).

Qualitative research refers to any research based on something that is impossible to accurately and precisely measure. For example, although you certainly can conduct a survey on job satisfaction and afterwards say that such-and-such percent of your respondents were very satisfied with their jobs, it is not possible to come up with an accurate, standard numerical scale to measure the level of job satisfaction precisely.

For more, check out this page from the University of Wisconsin-Madison Ebling Library guide on Qualitative vs Quantitative Research.