The original and still (as far as I know) most comprehensive discussion of this by Lincoln and Guba (1985). They introduced triangulation as a way to enhance the overall trustworthiness of qualitative research. Note that they prefer the term "trustworthiness" to validity.
I personally think that the most convincing use of triangulation is when two different data sources support the same conclusions.
The original and still (as far as I know) most comprehensive discussion of this by Lincoln and Guba (1985). They introduced triangulation as a way to enhance the overall trustworthiness of qualitative research. Note that they prefer the term "trustworthiness" to validity.
I personally think that the most convincing use of triangulation is when two different data sources support the same conclusions.
I fully agree with David. Triangulation of a single qualitative data set is more akin to 'comparing something with itself' and is often not compelling in terms of trustworthiness. If it was a multi-methods study, however, with more than one qualitative data set - then triangulation compares two different things - and is more compelling. Of course, it also depends on other factors - and is only an element of the four reecognised components of trustworthiness.
There are two ways at looking at triangulation. If using a qualitative case study approach the Yin recommends using several data sets to triangulate the findings. He would suggest more than three different data sets such as interviews, document analysis, observation and then looking at each. If the same findings are found in each then this makes the 'theme' more trustworthy.
There is another way to look at triangulation from an Interprevist view that states the more data we have from different sets then the richer the description can be as we can study a phenomenon from several different viewpoints. Thus, when we present the findings we can use 'thick, rich' descriptions which will allow the reader to determine if the findings are relevant to their context.
The aim is to strengthen knowledge collected and the ability to make good points for the discussion and answering the search questions. For this purpose of triangulation, we may use a questionnaire together with focus groups, for example. The responses in the focus groups will add more data and cover the limitations observed in the survey.
Sinead is right about the elements employed in trangulation. We also call corroboration. It is when we have different views collected from different sources on a similar question. For instance, an event in historical past is described based on data collected from different people directly knoweledgable about the matter or KI, fgd, written secondary sources would create what is referred to as trustworthiness of the matter
Most answers here only suggest "use 2-3 datasets". However, if one research does this, you may just be under involuntary confirmation bias because you see the same things in Dataset 2 than you did in Dataset 1 (i.e., you are primed). Therefore, ideally you would have 2 x 2 = 2 independent datasets, 2 independent coders. If the two datasets and the two coders agree, then you'd have a case for triangulation.