Petzval field curvature is an aberration that happens when a portion of light from a flat object is impossible to focus on to a flat image plane. Here, the object is where the light comes from, and the flat image plane refers to where we will see the image captured after passing through an optical system. Some systems, due to their curvature, have trouble focusing the flat object onto the flat plane. This is typically accounted for in camera lenses by adopting a curved, rather than flat, image plane.
Astigmatism is a term we hear often because it is a common trait of human eyes. As optical systems are in large part modeled after the eye, it is no surprise that optical systems also experience this same aberration. Specifically, an optical system featuring astigmatism means that it has two foci, causing objects and points off the axis fail to be sharply imaged.
Coma can be both a chromatic and a monochromatic aberration. As a monochromatic aberration, it refers to a distortion that might make objects appear to have a tail. This is a natural property of telescopes built with parabolic mirrors rather than spherical. The way light moves through the parabolic optical system can cause off-axis light source points to appear distorted. Specially tailored curvatures can mitigate the effects of this aberration.
These are just three of many monochromatic aberrations that camera lens designers must consider when building a functional camera or other optical system.