Apple's designated cloud photos solution is iCloud Photo Library using the Photos app on iOS and OS X. It stores and syncs your photos and videos in iCloud with one or multiple devices. It sounds like this should satisfy your requirements quite well (see the note below on iPhone storage).
- iOS 8.3 or later on your iPhone
- 10.10.3 (Yosemite ) or later on your Mac
- Sufficient iCloud storage to hold your entire photo library
Things to Note:
- iCloud Photo Library is not the same as Photo Stream. (What Photo Stream does exactly, is indeed not very clear. I would recommend just ignoring it or turning it off completely after you setup iCloud Photo Library.)
- iCloud Photo Library syncs both ways. Photos you take on your iPhone will also appear on your Mac (and all other devices that use it). Edits to photos are also synced.
- For each device you can choose whether to download the entire library or whether to optimize the device storage. On the devices where you choose the latter (your iPhone in this case), iCloud Photo Library will still take up some space. The space is managed internally and optimized for your device. It might keep things like thumbnails, favorite photos, photos you frequently or most recently accessed etc. Other photos are downloaded on demand. But the space used should be significantly less than keeping the entire library on the phone. Depending on how hard your "no space" requirement is, this might be a deal-breaker for you. I suggest you just try it out.
Setup iCloud Photo Library:
(Just to be safe, you might want to backup your photo library before proceeding.)
- Make sure you are signed in to the same iCloud account on both your iPhone and Mac.
- On your iPhone: Go to Settings > iCloud > Photos, then turn on iCloud Photo Library.
- In the same menu choose Optimize iPhone Storage.
- On your Mac: Open Photos (not iPhoto).
- Open the app's preferences (menu Photos > Preferences...) and under the iCloud tab select iCloud Photo Library.
- In the same menu choose Download Originals to this Mac. This will keep your entire photo library on that Mac.
Import your photos into Photos:
This step depends on your current setup. If you currently use iPhoto on your Mac, the photo library should migrate automatically. In general though, you can just drag and drop all of your photos and videos into Photos on your Mac.
Finally, let iCloud Photo Library upload and sync the library. This might take a while.