Termination, layoff, and non-renewal are 3 separate mechanisms through which an employer can stop working with an employee. The process for initiating each, documentation required, timeline, experience of the impacted employee, and perception of the employee by a future employer all vary. Below is a conceptual overview. For specifics to you and your university, you should consult with HR, and also read through the appropriate guidelines or handbook.

Termination

Termination is the HR term for "being fired". Generally it is "for cause" and the end result of a longer process involving documented Warnings and opportunities for the employee to improve, though it can be immediate.

Layoff

Layoffs occur when an employee is performing fine, but cannot continue to be employed. The most common reason for this is a lack of funding for their position, or the cessation of the program/effort/project that is the employee's primary focus. Layoffs also involve a documentation process with HR, but the focus is on the cause of the layoff, not on the employee. Most commonly there is a formal notification period, and the employee receives specific benefits.

Non-Renewal

A non-renewal is a simpler process wherein an employee working on a fixed term contract simply is not renewed when their contract expires. Sometimes advance notice is given, but not always. Different universities have different policies. The employee generally does not get any additional benefits as they signed the contract and thus agreed to the end date at the outset of their job.