1) The purpose of a gate review is to decide if a project is going to proceed to the next stage. In the APM lifecycle, they are held at the end of concept and Definition. They are typical chaired by the sponsor with the PM and other steering group members in attendance. They are primarily about the approval of the funding for the project. As such they authorise the project to process to the next stage. It is possible for the project to fail a gate in which case it will not proceed to the next stage.
2) Stage reviews are similar to progress reviews. They occur, mostly, during definition and the aim is to review progress against the plan. Unlike the gate review, the intention is for the project to continue. The review checks current progress and identifies any corrective actions. Typically chaired by the project manager and attended by the project team. The rely on KPIs of project performance such as milestone delivered in a month.
3) Peer reviews are conducted by people at the same level as the product owner. They can be technical reviews (of a design) or management reviews (of a plan). The aim is the get third-party input to the project. So, for example, you could arrange a peer review of your technical specification, procurement strategy or test plan. As such, they can occur anywhere in the project lifecycle. The aim is to decide if the planned approach is good enough for the project or can it be improved. They could form part of a quality control process in some organisations.
4) Benefit realisation reviews occur after the project output has been accepted. Typically, they occur at once the projects assets have been put into operation. They evaluate if the benefits defined in the business case are being delivered. The aim is to improve future projects, and focus the business on learning lessons. As such they are often conducted by the project sponsor, frequently with support from the project (programme or portfolio) office.
5) Audits are an external evaluation of compliance against a standard. They are conducted by someone outside the project, and sometimes from outside the organisation. They are typically about compliance and the identification of corrective actions. The reports are often shared outside the project team and can be in the public domain. For these reasons, they can less open and relaxed than the others types of reviews. However, they are a necessary part of any assurance programme