how can photographs be manipulated?

Using software. Double exposure (e.g. photograph one person on the left side of the photo and photograph a second person later so they look like they are together even though they were not).  Dodge and burn during the printing process to darken some areas and brighten others. . 

Any photo editing program will suffice. You can crop, change exposure, change saturation, rotate, sharpen, blur, add things, remove things like zits, warp things (make someone look thinner or fatter), keystone adjustments, add effects like film grain, reduce noise, exposure stack, focus stack, create panoramas, create anaglyphs (3D images when wearing red/cyan glasses), ....

There is a hash number that verifies that the image hasn't been altered. This system may have been discontinued due to the insecurity of the hash. This was used to prove that images weren't manipulated. Canon's & Nikon's image authentication has been hacked about a decade ago.

I think through photoshop. There are computer programs for this you can use.