Sometimes, it confuses me. What are under those concepts? Where should I classify action research, basic research, exploratory, causal or descriptive research?
Can you recommend a book or readings about those?
Research methods are means for collecting and analyzing data, while research designs have direct implications for collecting and analyzing data. All of the things you mention are more generic, and I personally would list them as "approaches." In particular, everything you mention could be considered an approach to doing qualitative research, so someone might be able to recommend an introductory text that lists a wide variety of different approaches to qualitative research.
To put it briefly, research approaches are the philosophical viewpoints that direct the study, research methods are the particular procedures used to gather and analyze data, and research design is the overarching plan or strategy that specifies how the research will be carried out.
As David L Morgan mentioned, research methods are means for collecting and analyzing data. The research design defines the type of research, population and statistical sample, and data collection tools. In general, it shows how to conduct research. I think the research approach refers to quantitative, qualitative or mixed research.
Research approach: what kind of research might you do to answer a particular question? Exploratory? Causal? Descriptive?
Research design: the plan or blueprint for your envisaged research.
Research methods: within your research design---the actual methods that you might use for collecting raw data for analysis, such as survey or unstructured interview.
For a quick answer, I think that Action Research is a bit of 'this' and 'that', perhaps most commonly, a combination of description and exploration since AS is commonly implemented when an 'old answer' in a particular field is perceived as now obsolete and requiring a 'new answer'---hence AS.