Assuming that this is everyday counselling, not some form of psychotherapy or cognitive therapy, the initial assessment may be very informal. However, some counsellors do use "assessment tools" that are used by a variety of psychologists, psychotherapists etc., and which are a series of questions designed to assess how well you are and to pinpoint your problems.
Some of the questions may seem a little intrusive, such as whether you have suicidal feelings or whether you have been sexually abused or whether you hear voices etc. It is entirely up to you whether you want to answer that sort of question.
Counselling helps a lot of people. If you find it helpful to talk to someone but find after a few sessions that either you can't talk to this particular counsellor comfortably or that you feel that you need a different sort of therapy, have a chat with your doctor about it.
It is quite normal for any sort of counselling or therapy session to last around 45-50 minutes. That allows the therapist to tidy up their paperwork and get ready for the next client within an hour, although for a first session, the counsellor will have more writing up to do.
Because I have severe mental illness, I have had a lot of therapy over the years. Whilst I have had therapy with highly trained doctors and therapists, I reckon the best therapy I ever had was common-or-garden counselling with a student counsellor.
Good luck.
