I was directed to this thread by another conversation and thought it would be best to post my response to the initial question before reading others responses so that I am not influenced by how others interpret and respond to the question... it should be interesting to compare things after that. 
Generic overview
I would divide my thinking into several basic common categories. In general, a lot of my thinking involves asking myself questions and either trying to answer or respond to those or see what other questioning branches those lead to - when I have reached the end of a basic chain of thought, I start sub-consciously looking for tangent thoughts to move to next.
Specifics - Interacting with someone I already know.
Is there a specific reason for the interaction?
Has anything important changed since I last interacted with them?
Do they want or need anything from me?
Do I have things I want or need to talk to them about?
Are there recent events regarding shared interests that can be discussed?
Specifics - Evaluation of another person.
Get a rough impression of them based on intuition and past knowledge
(this is usually a sub-conscious process, but I theorize that Se, Ne and Si are working together to gather data, reference it against past experiences, and make a quick evaluation)
Over time, I try to stay alert for subtle things that will tell me things about their value system, their interests and their motivations. Once I feel I have a decent picture of the person as a whole, I can determine if they are someone I might enjoy talking to, someone to avoid, or someone I'm simply indifferent toward.
Specifics - Evaluation of an object.
Ask what its intended use is.
Ask myself if I have any immediate use for it.
Ask myself what alternative uses I can think of for it.
Specifics - Evaluation of a value.
Since I already have a large mental database of existing values, the first step is to compare it to those and see if if can fit in the existing system somewhere. If its something I find myself opposed to on the basis of existing values, I ask what value it conflict with and what about it bothers me. I store the conclusions in my mental database. If I encounter a situation that can put the value in a new light or show it from a different perspective, I go back and re-evaluate it asking things like "when would this be acceptable, and when is it unacceptable" and "what about this alternative scenario makes the value more or less acceptable" and things of that nature.
Specifics - Evaluation of an idea.
When I get a new idea, I spend a lot of time looking at all the possible effects and figuring out what aspects of it I have problems with and then look at what redeeming values the idea has and weighing those against each other (since ideas usually involve value judgments of some kind, I filter the values inherent in the idea through my value system to look for things about it that are disagreeable and what things about it fit well with my values). After I feel a have a good grasp of the pros and cons of the idea I start looking for ways it could be improved or approached differently to mitigate the cons or enhance the positives. It is very common for me to have a large reservoir of ideas stored away and on occasion I pull one out and play with it tweaking the variables and such and just playing with all kinds of different "what if" scenarios. Most of my ideas are simply impractical, cost prohibitive or too idealistic to try to implement, but its still enjoyable to play with ideas and sometimes there is even a chance to try to incorporate one into daily life.
Specifics - Problem Solving
I ask myself what my concern is.
I ask myself what my options are.
I run through mental scenarios to evaluate the probable outcome of the various options.
I narrow those down to several of the most probable, the most desirable and the worst case scenarios.
I try to determine how plausible the desired result is and if there are multiple ways to reach that result.
- if there are multiple options with similar favorable results, I flag the less conventional approaches as preferable (even if its at a slight cost of efficiency or would need more effort)
I start asking what things I can effect to further increase the probability of favorable result.
I try to look for important things not to do to avoid causing problems.
I store those scenarios and variables in a mental filing system to have them available so I will be prepared when needed.
Edit/Append:
For a few more generalizations, a lot of my internal thoughts as I mentioend is question/conversation based - I ask questions until I think I've covered as much as I can think of about something. Sometimes I'm just talking to myself, but sometimes its like I'm in a virtual world talking to other specific people to bounce ideas off them (with my mind filling in what it thinks they would say/do). When I'm playing with an idea that involves physical aspects of the world (a lot of my ideas are emotional or sociatal in nature and the physical world need never be considered for their aplication), its a bit like I'm playing with a giant sandbox or VR erector set of somethign like that where I can add and remove things from the environment and see how that effects things. Yet another way to look at it would to be see myself as playing in a dreamworld where I can control time and make any changes I want and watch the effects of those changes from any angle I want and conjur others to talk to as way to try to figure out how things work or what the effects of various actions might be.
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