Agile and waterfall are two distinctive methodologies of processes to complete projects or work items. Agile is an iterative methodology that incorporates a cyclic and collaborative process. Waterfall is a sequential methodology that can also be collaborative, but tasks are generally handled in a more linear process.
Following the agile methodology, your project will move through a series of cycles throughout the lifetime of the project. The development phase, review, feedback, and then approval of the work item – either yes or no. If yes, implement and complete the task. If no, record and make any necessary changes, track and adjust the backlog or prioritization to reflect the newly acquired knowledge, and then move onto the next task or sprint.
Following the waterfall methodology is a simpler process of moving tasks through the phases of defining requirements, designing the implementation, implementing the work item, verification of implementation and quality assurance, and then maintenance of the feature in the end.
Selecting the right methodology for your projects will depend on preference and the nature of each project. Some projects require a more iterative process and others require a more sequential approach.
Here are some other articles on the topic, you might enjoy: