Apple's iCloud Photo Sharing allows for sharing photos between Macs and iOS devices. This can come in handy for syncing photos across devices; like taking photos with an iPhone and viewing on the larger screen of an iPad.
Photos only stay in a personal Photo Stream for 30 days, but even then not everything necessarily needs to get shared in a Photo Stream. For example, if you take multiple photos to make sure nobody is blinking, you may not need to keep all of them. Perhaps you're, say, taking a bunch of nude selfies, you may not want those in Photo Stream either. If that's the case, you may want to disable Photo Stream entirely.
Here is how to delete photos from My Photo Stream:
Using iOS 7
Go to Photos, tap Albums, then tap My Photo Stream.
In the list of photo albums, choose My Photo Stream
To delete one, tap the photo and tap the trash icon in the lower bar, then Delete Photo.
To delete several, tap Select, tap photos to remove, then tap the trash can  and tap Delete X Photos.
Choosing which photos to delete from Photo Stream.
On an iOS 6 device
- Go to Photos, then Photo Stream, then My Photo Stream.
- To delete one, tap the photo and tap the trash icon in the lower bar, then Delete Photo.
- To delete several, tap Edit, then tap the ones you want to remove, tap the trash can, and tap Delete X Photos.
- Tap Edit, then tap one or more photos.
- Tap , then tap Delete Photo or Delete Selected Photos.
On Apple TV
- Go to iCloud Photos.
- Select My Photo Stream, press the center button on your remote, then navigate to a photo you want to delete.
- Hold down the center button on your remote, then choose Delete Photo.
- NOTE: If you simply press the center button instead of holding it down, a full-screen image appears instead of the dialog where you can remove the image.
iOS 8
In the iOS 8 beta, things look the same as they do in iOS 7 when it comes to Photo Stream.