**** I am not a teacher ****
To me, a job opening and a vacancy means the same thing. Usually a job opening is advertised online or in newspapers. The job openings may also be found on some companies' websites. When a candidate sees an advertisement for a job opening/vacancy and applies for the job, s/he is interviewed, and, if found suitable for the job, a job offer is made to the candidate. The job offer letter basically tells the candidate that the company is offering him a job, the title of the position they are offering, the starting date, and the salary. It also usually has a date mentioned by which the candidate must accept the job, if interested.
Please note that not all job offer letters will have the information mentioned above. What kind of information is in the job offer letter may depend on the kind of job. The job offer letter is also known as job appointment letter in some places.