You can think of participant observation in design as a fly-on-the-wall research method. The idea here is to blend into a community and observe from the inside. This reveals behaviors, attitudes, experiences and their causes that the community members are often not even aware of themselves. The balance between participation and observation obviously varies with context. There is always a danger that the presence of an observer alters the conditions of observation.
George Plimpton is probably the most famous participant observer. As a writer he played with several professional sports teams in real games (for example he pitched again Willie Mayes, sat in goal for the Boston Bruins, and boxed Sugar Ray Robinson all for the purpose of writing articles or books).
Like participant observation, contextual inquiry also relies on deep immersion in the community of study, but it follows a master and apprentice model. In a contextual inquiry the designer works in partnership with a subject matter expert in their environment to learn how they do, see and experience things.
There is very little structured questioning in a contextual inquiry, and the success of this method relies on the genuine curiosity of the designer, and his or her ability to play to role of apprentice. A frequent danger in this method is there is pressure on the subject matter expert to show how things are supposed to be done rather than how they are actually done.
Roughly speaking, ethnography, a specialization of cultural anthropology, is the qualitative field study of the complex interdependent elements that make up a given social fabric, activity and experience.
Of course design is not anthropology, so we needn't cast such a wide net. We focus on the narrow social fabric, activity and experience related to your products and your customers.
It is very important to note that ethnographic interviews are neither about soliciting feedback on current ideas nor are they about asking customers what they want. Ethnographic interview in design are about discovering and expanding knowledge and insight to better understand the problem space and then drive more innovative and more appropriate solutions.
Depending on how much learning has taken place prior to an ethnographic interview, the interview's purpose can range between opportunistically exploring a problem space to validating prior learning. Regardless of purpose, the idea is to partner with interviewees, and help them open up and share their subject experiences and knowledge, while subtly keeping their feedback focused on the opportunity rather than evaluating or judging solutions.