To analyze: consider in detail and subject to an analysis in order to discover essential features or meaning (WordNet).
Studying people, places, speeches, text, and things in general is a very important process of everyday life. You become familiar with what you are looking at, you know its ins and outs -- all of its intricacies. Yet, there is a fine line between analyzing something and over-analyzing something. Once you put one toe over that line, you've stepped into very dangerous (and potentially harmful) territory.
When one over-analyzes something, one is simply finding things that just aren't there. This is where the danger rests; when you begin to find something that doesn't exist, you are fooling yourself into believing a lie.
A good example of this (something that I often do) is to over-analyze a short, written message, such as a text message. Sure, sometimes there is intonation, and surrounding context is also often a factor, but much of the time, a considerable amount of emotional information is lost in translation. Text messages reduce a person's ideas, emotions, and intentions into 160 short characters. To analyze such a message in depth would be meaningless, unless the sender has a particularly cryptic personality. Yet, I do it, and I'm sure many, many other people do as well.
With verbal or in-person communication, one can get considerably more information as words are spoken. Tone of voice, eye movement, facial expressions -- all of these, along with speech, combine to form an idea and intention. Of course, it is still possible to over-analyze even this amount of information (especially short conversations, though sometimes those can be the most telling), but it is far less likely.
The best thing to do, in my opinion, is to take things at face-value. Feel free to consider whatever it is with a bit of insight, but don't obsess or over-think it. Likely, what you are looking for is right in front of you...or not there at all.