They all typically involve assigning a value, whether cash value or some other value (such as emotional value) Estimate can be used to make a guess on any value. Estimate and guess are interchangeable, but estimate implies a more correct or better thought out guess. For instance, the scientist estimates there are one hundred billion stars in this galaxy. Here the scientist is assigning a value to the number of stars. Appraise is used almost exclusively with cash value. For instance, he appraised the house. Here he evaluates the house and gives it a cash value. Assess and evaluate have almost the exact same meaning and are usually interchangeable (examine is interchangeable as well). This is to consider or look at something. They typically don't involve cash value. You may evaluate a situation. You may assess the quality of penmanship.

I suggest that you look up the meanings in www.thefreedictionary.com