Since Apple announced that the new iPhone would have a fingerprint reader there has been a lot of concerns about privacy. I'll try to explain the fingerprint technology that Apple is using as I understand it and attempt to address some of the concerns I have heard being raised.
What I present here is based on my experience as a developer working with the Authentec SDK (whom Apple acquired a little while ago) but I am by no means an expert. My experience is limited to the Authentec SDK and the Zvetco fingerprint sensor.
A high level look at what is happening
When you place your fingerprint on the sensor, it doesn't "scan" your fingerprint like your typical scanner. The sensor maps the electrical conductivity of the fingerprint and generates a model of your fingerprint. This can then be converted into an image which would resemble your fingerprint, but it's important to understand that it's done via conductivity because a paper is not electrically conductive and thus you could not trick the sensor with a picture of a fingerprint.
Once the fingerprint has been mapped by the sensor the next step is to generate the fingerprint template. The template is basically just enough information about your fingerprint to match it again, but it is not an image of the fingerprint. This is what Apple will store, and I do not believe that you would be able to recreate the fingerprint image from the template.
The fingerprint image
As I stated above, the map of the fingerprint can be visualized in a what that very closely resembles your fingerprint because it is almost like taking a picture of your finger. The lifetime of this map, however, is likely very short. Typically once the fingerprint has been scanned the sensor notifies the operating system that a new map is ready and the operating system would then pass that on to the authentication system which will request the fingerprint map data from the sensor.
Once the sensor provides the data to the requestor, it could immediately discard the information or after a certain period of time. I cannot tell you for certain what the iPhone sensor does, but I would guess that if it doesn't discard immediately, it does so within a short period of time.
The template
When it comes to the actual information the template actually stores, I'm a complete novice. I'm going to speculate on what it likely stores, but I would definitely not consider it to be correct, but likely something like that.
The template represents enough information to repeatedly match the fingerprint, but it is not a monolithic piece of data. Typically, you will generate the template by "enrolling" the fingerprint which consists of three to five scans of the fingerprint. Once the enrollment is done, you can now match subsequent fingerprint scans to the template. Typically, you would update the template with subsequent scans as well, improving the accuracy of the matching over time.
I have seen other articles which state that perhaps Apple is hashing the template, which would make the template indecipherable, but it would also make the accuracy very low and would not allow the template to be updateable, so I do not believe that is what Apple is doing. At the same time, I do not believe that you can get much information about the fingerprint from the template.
So what is in the template? Like I said, I really don't know. If I were to guess, I would say that they first detect some of the more prominent fingerprint features (such as swirls, loops, etc), and then take some measurements between those features. The algorithm should pick the prominent features each time, and thus is just needs to validate that the measurements are within tolerances and then it would be considered a match.
If you are curious as to what data the template stores, you can download my fingerprint template and play around with the data and see what information you can extract from it. This template was generated using the Authentec SDK about 3-4 years ago. I would be genuinely interested in what you can decipher from the template.
Conclusion
Ultimately, I don't think there are very many privacy concerns here. I don't believe you can reverse engineer the template back into something that resembles a fingerprint, which is why I feel comfortable posting my template onto the internet. Apple claims that the template is not transfered anywhere, so the privacy concern should be even less.