How to fix red eye or pet eye in a photo
The Red Eye tool in Paint by Zaweba sets pupil size and darkening, and its Eye: Animal option removes green, yellow or white eye-shine from pets.

Pick the Red Eye tool and click on each pupil. In Paint by Zaweba the tool has a pupil size setting and a darken setting, and for cats and dogs there's an Eye: Animal option that removes the green, yellow or white eye-shine animals get from a flash, with an optional catchlight so the eye still looks alive.
Red eye happens when a flash close to the lens lights up the back of the eye. In people it comes out red. Animals have a reflective layer behind the retina, which is why their eyes glow green, yellow or white instead. The two need slightly different fixes, which is why the tool has a separate setting for them.
What you need
Only the photo and Paint by Zaweba on Windows. It's free, offline and needs no account, so this is a quick job you can do on any snapshot.
Fixing red eye step by step
Open the photo and zoom in on the face so the eyes fill a good part of the screen. Choose the Red Eye tool. Set the pupil size to roughly match the pupils in the picture, and the darken amount to how dark you want the result.
Click on the centre of each red pupil. The red is replaced with a natural dark tone. If the result looks too grey, raise the darken setting and try again. If a red rim is left round the edge, undo and try a slightly larger pupil size.
Do one eye, check it, then do the other with the same settings, so the two match. Mismatched eyes are more noticeable than a little leftover red.
Fixing pet eye
For animals, switch to Eye: Animal. Click on the glowing pupil and the green, yellow or white shine is removed. Because an animal's eye can look flat and lifeless with the glow simply blacked out, the option can add a catchlight, the small bright point of reflected light that makes an eye look wet and alive. Leave it on unless the eye already has a natural highlight.
Animal pupils come in different shapes, so look closely after each click. A cat's slit pupil in bright light needs a smaller setting than a dog's round one.
Getting a good result
Keep the fix inside the iris. If the tool spills onto the coloured part of the eye or the eyelid, reduce the pupil size. Keep any real catchlight that is already there, because it is what makes eyes look natural. Zoom back out to normal viewing size before deciding you are done, as tiny flaws at 400 percent rarely show in the finished photo.
When the tool isn't enough
Now and then a pupil is too damaged by the flash to fix with one click, with the colour bleeding into the iris. Then it is worth a manual repair: select the pupil, darken and desaturate it with an adjustment layer limited to the selection, and paint a small soft catchlight on a new layer. Paint by Zaweba's healing tools can tidy any stray colour round the edge.
Avoiding it next time
Red eye and pet eye come from the flash being close to the lens. Bouncing the flash off a ceiling, using more room light, or having the subject look slightly away from the camera all reduce it. When it does happen, though, a couple of clicks in Paint by Zaweba is usually all it takes.