With Photo Stream enabled in iCloud settings on an iDevice, a copy of the photo you just snapped will be automatically uploaded to Photo Stream. You can then access the photos in Photo Stream folder from any devices including the Mac.
iCloud will store only the last 30 days of photos. Older photos will be removed automatically by iCloud. There is no way for you to manually remove individual photo in Photo Stream. You can however reset your Photo Stream which will remove all your Photo Stream photos from iCloud.
To reset Photo Stream, you need a browser. Go to icloud.com website and sign in using your Apple ID.
Click on your account name at the top right corner of the page.
On the Account setting screen, click Advanced, and then click Reset Photo Stream. Click Done to close the Account screen.