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Correcting in software straightens lines but has to crop to fill the frame.
What lens distortion is
A perfect rectilinear lens draws every straight line in the scene as a straight line in the picture. Real lenses magnify the edges of the frame slightly more or less than the centre, so straight lines near the edges bend.
Barrel distortion
Magnification falls toward the edges, so straight lines bow outward like the sides of a barrel. Typical of wide angles and the wide end of zooms.
Pincushion distortion
Magnification rises toward the edges, so lines bow inward toward the centre. Typical of telephoto zooms at their long end.
Moustache distortion
A mix: barrel near the centre turning into pincushion at the edges, so a straight line waves like a moustache. Common in compact ultra-wide designs and hard to correct by eye.
Measuring it
Distortion is quoted as a percentage: how far a straight edge line is displaced relative to the frame height. Under about 1% is rarely noticed; 3% and up shows on horizons and architecture.
Software correction
Lens profiles in cameras and editing apps warp the picture back into straight lines. Because the corrected shape no longer fills a rectangle, the result is cropped and slightly softened at the edges.
Why cinema cares
Visual effects work needs to match real footage to computer-generated scenes, so cinema lenses are often designed for low distortion, and every lens is mapped with distortion grids before VFX shots.