Kanban vs Kanban
Kanban originated in Toyota where it was (and still is) used to organize processes in their factories, such that they can quickly respond to changing demand and keep the production running smoothly. Wikipedia has a good overview of Kanban as part of the Toyota Production System.
Kanban in software development is one of the Agile processes that aims to achieve the same for the dev environment. In contrast to Scrum (the most popular one), it doesn't have set time cycles (sprints) and is considered more lightweight (I'm simplifying a lot here).
Here's the difference between what goes on in a software shop vs a car factory:
If a factory is producing N widgets (car parts), every widget is the same (mass production). If a software teams are producing deliverables (software that gets deployed), each piece of software is different (otherwise we'd just copy-paste it).
Kanban (Toyota) is an method of sending purchase orders (ie. requests to make widgets of specific type in some quantity) from the team(s) who need them to the teams who make them.
In software, quantity is always 1 and while the “widgets” have notionally been designed (specced out in the ticket), they have never been built before, unlike in the factory where every widget at least had a test run before.
That is, Kanban as used software development is only loosely inspired by Toyota, and the essential difference (time to reconfigure the assembly line for another widget, vs time to design, prototype and test the widget) doesn't carry over.
In software, the main benefit of Kanban is managing the development load. Developers pull work items (tickets) from the backlog when available, and if blocked, can take another ticket until the work-in-progress (WIP) limit is reached. This provides clear visibility into the project progress, allows for reprioritization, and minimizes work interruption.
Combined with the simiplicity of Kanban when compared to Scrum or other agile methods, it's easy to see why Kanban is a popular choice for software development teams.