When do straight lines become curved when talking about projection?

When you decide what type of projection you wish to use.

A rectilinear projection will preserve straight lines (but not correct angles between straight lines) at the expense of the relative sizes of objects in the center of the frame versus the edge of the frame. Things near the edge will be distorted in size to preserve straight lines. This works best with photographic lenses when the angle of view is relatively narrow. The wider the field of view, the more distorted the angles between straight lines become and the more distorted the relative sizes of things in the center and things on the edges becomes.

A fisheye projection will preserve relative sizes at the expense of not preserving straight lines.

An orthographic projection will preserve both straight lines and the angles between them. To capture an orthographic projection with a camera requires a telecentric lens that is larger in diameter than the width/height of the scene being projected. Many architectural drawings ('side view', 'top view', etc.) use orthographic projection. Each point in the scene is drawn from a constant angle rather than from the perspective of a constant point of view. Scanners and copiers capture an orthographic view of flat documents by travelling across the face of the document as they image each part.

There are many other types of projections, but these are the most common seen within the context of photographic lenses.

In the photographic disciplines (which is the specifically expressed subject of this site) the term projection is used when discussing how a lens projects a 2D image of the 3D world onto a flat imaging plane. In the disciplines related to artistic drawing, the term perspective is used in much the same way. But in the photographic disciplines the term perspective refers not to how an image is projected by a lens, but rather the relative positions of the objects within the camera's angle of view and how the relative distances from the camera to each of those objects affects their relative shapes and sizes as well as what part of a more distant object may be hidden by a closer object. Changing projection will alter the shapes of things, but not what parts are hidden by closer objects. Only changing the camera position, that is the perspective, can change that.

For more, please see:
What is the difference between perspective distortion and barrel or pincushion distortion?
Does wide angle equivalent in crop sensor skew image?
How do i take a photo of an object perfectly vertical like a top view?
Why the faces in the corner tends to skew a bit in almost all smartphone cameras
How wide can a lens be before it begins to produce distortion?
What does it really mean that telephoto lenses "flatten" scenes?
What's the opposite of a fisheye lens?
How can I achieve an effect in still photography that's analogous to a cinematographic effect?