As far as I can tell, morphological analysis is a technique used by Bible students to better understand the original languages in context.
How does one go about using a tool such as Robinson's Morphological Analysis Codes?
Most of the answers so far are getting all hung up over very specific examples of morphological analysis. It is by far better to start with the basic notion of "morphological analysis", then look at how each of these instances already mentioned meet the goals of morphological analysis.
Morphological analysis means exactly what the etymology suggests: it is analysis of the forms of the words used, explaining their grammatical significance. The obvious example is picking a word in the text, such as δεδικαίωται, looking at its form to not only tell what verb it is from (in this case δικαιόω), but how the endings (rarer: prefixes/infixes), modify the basic meaning of the word.
In this case, that means explaining that it is perfect passive indicative third person singular of the verb δικαιόω.
The analysis may or may not include how the general rule for forming the specific form applies in the specific case. So, for example, it may explain that the δε prefix marks the perfect tense, and that the ω in δεδικαίωται is the result of contraction with a standard marker for the third person in secondary tenses, a historical σ being dropped. But this kind of detail is not common for the tools offered by sites like Blueletterbible.org or eSword. Perseus gets close, though.
To really familiarize yourself with what a morphological analysis tool does, I would start with the one at Perseus rather than either Blueletterbible or Robinson's codes. This is because the Perseus tool gives a much more complete analysis, also including the ambiguities that these others have resolved by relying on more than mere morphology. The main drawback to the Perseus engine is that they removed the feature of allowing entry of a form to analyze via Greek keyboard: you now MUST enter it using their Betacode, which I find unpleasant. But it is not that hard to learn.
A simple way to explain the RMAC (Robinson's Morphological Analysis Codes) is that words in the Greek language changed ('morph'-ed) their form depending on how they were being used (and therefore to be understood) in a given context. We do this same thing with many of our own words: give, given, giver, gave; have, has, hasn't, had; go, going, gone, went, etc. What Robinson did was he developed a simple, concise, Alpha-Numeric code system to accurately define, grammatically, how words were to be understood in the Bible as regards their 'parts-of-speech': whether they are a noun, verb, pronoun, adjective, preposition, participle, etc.; as well as their person (1st [I/me], 2nd [you], 3rd [they]); as well as other basic grammatical features such as their tense (past, present, future); mood; case; gender; etc. To understand the scope of the problem (and the need for such a code), consider that Greek verbs alone, could 'morph', I believe (my Greek is from 30 years back [and rusty]), into more than 100 different endings/forms; some of which were identical for different applications of the same verb; and their interpretation/application must be determined by understanding their morphology, by the rules of Greek grammar, and by context. RMAC contributes greatly to clarifying these issues. R. Summers' "Essentials Of New Testament Greek" has an excellent fold out of these endings/"morphings" of Greek words (I wish I still had his book). Robinson's codes are based on the accepted rules of Greek grammar, and apply to the entire Greek Old Testament (I believe), as well as the entire New Testament. The RMAC eliminates much 'personal' interpretation of the Greek Scriptures by accurately defining how the Scriptures should be understood morphologically/grammatically, according to the accepted rules of Greek grammar; and is, therefore, a very useful tool for aiding in accurate Biblical interpretation. To understand, appreciate, and apply RMAC one must have at least a fair understanding of English grammar as shown briefly above. e-Sword has a very good (and easy to use) application of the RMAC, and used to be free; but now e-Sword is fee based. I am not personally familiar with other RMAC availability options, some which can be found on Google or on Bing.
This is far from a complete answer, but I thought I'd add a few notes on what I've found so far to potentially help anyone else who tries to figure out what morphological analysis in bible study is for, until we get some more answers.
I found the term "morphological analysis" when I was re-installing e-Sword and noticed a plugin tool called "Robinson's Morphological Analysis Codes."
That's all I know so far. I'd still like to hear from anyone who is familiar with morphological analysis. My "word study" skills are about as basic as it gets, and include looking up a Strong's number and checking a lexicon. I'd like to know more about what kind of insights can be gained through more thorough analysis of the original languages. Cheers.