How can I cut an arc into a square plane? The gif above shows what I want to do, but, in another software.
You can achieve this with the Bevel modifier.
Tab into edit mode and select the corners you want to curve. Press CtrlG followed by A or click Mesh -> Vertices -> Vertex Groups -> Assign to New Group to assign them to a new vertex group.
Tab back to object mode and add a Bevel modifier.
Enable Only Vertices set Limit Method to Vertex Group and select the vertex group you just created (by default it will be named Group). Adjust Segments and Width to meet your needs.