Ancient Greeks and Chinese.
No one person can take the credit. It was a long process of many different people developing techniques and understanding how light worked, and how it might be captured in an instant image. Some of the basic milestones are:
~450 BC : Basic principles of optics and a rudimentary camera idea worked out by ancient Greeks and Chinese.
1665: Isaac Newton showed that light was separated into different colours.
1727: Johann Heinrich Schulze found silver nitrate darkened when exposed to light.
1814: Joseph Nic�phore Ni�pce makes photographic image with camera obscura - this faded quickly & took eight hours of exposure.
1837: Louis-Jacques-Mand� Daguerre's shows off his first daguerreotype (only a handful survive today). It didn't fade, but still needed 30 minutes exposure.
1841: William Henry Talbot presents the Calotype process - allowing multiple copies of pictures. Talbot is often accredited with inventing the photography process as we know it, using negatives to make positives (actual photographs).
1851: Frederick Scott Archer derives Collodion process. Now images could be made after only two or three seconds of exposure.
1913: First 35mm camera developed.