Aberrations, or optical imperfections, exist because no lens is optically perfect. Almost all lenses show aberrations from the ‘perfect’ image.
Distortion is a general term for when lenses fail to show straight lines as straight and instead show them as curve or bowed. You can get barrel distortion, where straight lines appear to bow outwards towards the edges of the picture, and pincushion distortion where they bow inwards. Fisheye distortion is an extreme example with extreme wideangle lenses where no attempt is made to correct the distortion.