Learn English – Difference between “picture” and “photo”

A "photo" is something that was taken with a camera.

Short for "Photograph". Photo means 'light'.

An "image" is something generated or a representation of something in any other way. This can be a combination of different photos, part of a photo that has been greatly edited, or really, anything that has been made in Adobe Photoshop.

All elephants are gray things, but not all grey things are elephants. All photos are images, but not all images are photos.

"Category" and "type" are largely synonymous with regards to their similar meanings. ("Type" obviously has a few other uses.)

The only real difference between the two is that a category feels more like a bucket or classification: Things are filed or stored away in categories. A type is more akin to a label or identification.

My type of man is tall, dark and handsome.

Place this in the loser category.

In both of these examples, you could not switch the words out for the other. This isn't so much a grammatical requirement; people just don't do it.

Also of note, the terms have become absorbed by Computer Science and Mathematics with very specific meanings and purposes. For more information, see the relevant Wikipedia pages for Type and Category.