An assessment is a process of rating or judging something against a concrete set of standards.
For example, if you are assessing a patient you may be taking readings such as blood pressure and respiration rate, and making a judgment on the state of the patient based on the results.
An estimation is a reasonable guess or opinion.
For example, if someone hands you a box of marbles and asks you how many are in there, without counting them, you would say something like 'In my estimation, the box contains 50 marbles.'