Okay, let's see if I can tackle this...
Almost all of my answer to this question is being drawn from the first chapter of Michael Posner's Foundations of Cognitive Science. I, myself, am just a rising freshman at a decently ranked American university who has been studying the cognitive science literature on my own for the last few years.
Here I will attempt to explain one (and perhaps the most significant) example of introspections in modern cognitive science: protocol analysis.
I've only recently learned about it, but with the book here in front of me, I'm sure I can manage to give a good summary of it.
Protocol analysis, simply put, is the use of verbal reports of mental activity as data. It's not quite the same as the introspection of early psychology. From Posner, "Although verbal reports date back to the introspectionists, the use of such reports as data should not be misconstrued as introspection. Introspection took subjects' verbalizations at face value as constituting a valid theory of their own thought processes. Protocol analysis today, however, treats verbal reports as a source of data to be accounted for with an experimenter-generated theory, perhaps in the form of a computer simulation.”
The instructions given to subjects about how to report their introspections seems to have a great impact on the content of verbal reports. Here are some examples:
- You could ask someone to "Think aloud" as they are performing some task. ("25 plus 25. 2 plus 2 is 4, and 5 plus 5 is 10, so carry the one and the total is 50.")
- You could ask them to report their mental processes as they are performing their tasks. These tend to elicit more theoretical answers ("I am now visually processing..."), especially since you're probably questioning college psychology students. These answers are usually not preferred by modern psychologists.
- You could ask the subjects questions. ("How did you decide to do that?")
And so forth. The verbal reports get transcript-ed and then the experimenter then segments the report into parts that can be encoded (translated into a formal language, like mathematics) independently. Reliability of the data becomes higher the more simple and precise your coding scheme is. In fact, some systems have been invented for the automatic coding of segmented protocols, though they don't rival human experts quite yet.
After analyzing many subjects' narratives in this way, you have loads of rich data about how people think through a given task. Since we aren't introspectionists, of course, we don't interpret this data at face value. So how do we interpret it?
“Verbal protocols generally provide explicit information about the knowledge and information heeded in solving a problem rather than about the processes used. Consequently it is usually necessary to infer the processes from the verbal reports of information heeded instead of attempting to code processes directly.”
So protocol analysis is most useful for understanding the knowledge consulted when one is thinking through a task. The verbs that participants use to describe their thoughts often provide useful clues about the processes involved in completing a task. For example, the tenses of verbs can distinguish between the retrieval of information from memory and the planning of future actions.
Since we are cognitive scientists, after all, our interpretation of verbal reports are usually informed by information-processing theories of thinking aloud. In other words, our science of introspection improves our ability to derive correct conclusions from introspective data.
But yeah. As a tool for guiding the creation of more precise hypothesis of human thinking (with computational models), introspection is still quite useful. Today it's more diffident and rigorous and better informed, but still alive.
You also can't understate the role introspection probably informally plays in driving the hypothesizing of cognitive scientists. This has less rigor, however, and may even be holding us back.
I keep saying "us" like I'm a scientist already. I'm not. >_<
