Locate the thesis of the argument you are analyzing. The author or presenter will often state it in one succinct sentence close to the beginning of the article, essay or presentation.
List each argument and piece of evidence in support of the thesis and leave space for notations.
Analyze the logic, facts and any data that the argument presents. Look out for emotional arguments, hasty generalizations, and red herrings, which a sound argument must omit. Also look for erroneous facts, omissions of facts that you know should be there, and data that is dated or taken out of context. Make notes as you work.
Look at studies that the author quotes if they seem suspect. Sometimes researchers do only short studies or studies that do not include a large enough sample. Sometimes they don't ask the right questions or the methodology is weak. Also, the references should come from credible sources; credible sources are those written by research scholars in the field or practicing experts. Make notes as you work.
Open your analysis with a paragraph that ends with your own thesis, either agreeing or disagreeing with the other person's thesis.
Address the argument point by point. Do so in the same order in which the author or speaker presented his points. Alternatively, you can group related points together. Concede valid points, but point out flaws in others. Save your strongest, most important point for last.
Weave in concrete support for your analysis. Cite reliable, current references.
Conclude the analysis with the discussion of your strongest point or with a short discussion of the subject matter as it pertains to your thesis.