How Photo Stream Works
If you have Photo Stream enabled on both iPhone and iPhoto, any picture you take on iPhone is automatically uploaded to Photo Stream, and the automatically downloaded to iPhoto. iPhoto must be open for this to occur (but don't worry, the photos stay in Photo Stream for 30 days or until 1000 more pictures are taken, whichever comes first, so you don't have to have iPhoto running constantly).
Uploading Photos to Photo Stream
Photo Stream only uploads from iPhone while connected to WiFi. Occasionally a picture is missed, and as of now (August 2013) videos are never uploaded. There is no way to trigger an upload later if a photo fails to upload to Photo Stream. You can, however manually sync iPhone with iPhoto to sync videos as well as any missed photos.
Syncing iPhone with iPhoto Manually
Plug iPhone into your Mac then in iPhoto, click on your device in the left sidebar. There will be an Import button. It will automatically skip any photos previously imported from Photo Stream. Once imported you can choose whether to remove them from Camera Roll on the phone. If you do this, make sure you are syncing them back to iPhone in iTunes if you want them on the phone in albums.
Deleting from Photo Stream
When you delete a photo from Photo Stream, it removes it from Photo Stream entirely. If it was already downloaded into iPhoto it will remain there under the (MonthName Year) Photo Stream album, but not in Photo Stream itself.
Deleting from Camera Roll
When you delete a photo from Camera Roll it removes it from that area but does not delete it from Photo Stream.