You can only delete photos that are in the Camera Roll, or Imported Albums. Photos that were synced using iTunes can only be removed by changing the sync settings in iTunes (or contents of the synced folders), and syncing again.
The Remove option appears when you have added a photo to an album locally on the iPad. Since this only links the file to the album (the original picture stays in the Camera Roll), you are only removing the photo from that album. Not deleting it.
If you try to delete a photo from the Camera Roll that has been added to an album, you will receive a warning that you are going to delete that photo from all albums, as well as the Camera Roll.
The other icons (like Red Eye) appear when you are editing a photo. You can not delete the photo when in Edit mode, only cancel the edits, or save.