Restricted Supercharger use will apply to any new Tesla ordered after January 15, 2017. (So you still have a few days to get an order in if you want to be grandfathered in to the original unlimited plan.) Owners of cars ordered before then will still be able to charge as much as they want, and when the vehicle is sold the unlimited charging will go with it. Owners of newer Teslas will get 400 kWh of charging annually, enough to drive about 1,000 miles, which will not roll over from year to year and will instead reset on the anniversary of the vehicle's delivery date. If the car is sold, the total will be reset on the date of transfer and that will become the new annual reset date.
The cost structure will vary by location – state to state in the US or country to country elsewhere. Owners will be charged either by the kW (in most locations) or by the minute (because some local regulations require it), and Tesla says it's working with lawmakers to get the per-kW charging system in more places. The charge fees vary in part due to local energy costs. Charging fees will be charged to the credit (charge?) card on file on Tesla's site, where all of this will be managed.
Tesla splits the per-minute rates up into tiers to make it correlate somewhat to what you're getting when the charging rate changes, with Tier 1 charging applying when the car is charging at a rate of 60 kW or less and Tier 2 is for when it's charging above 60 kW. The charge rate depends on how full the battery is, whether you're sharing a Supercharger with another car, and other conditions like the weather. As an example, in our home state of Michigan the pricing for Tier 2 charging is currently set at $0.20, while Tier 1 is $0.10. (Tier 2 appears to generally be twice the cost of Tier 1.) In the state of New York, charging is per kWh and set at $0.19. Tesla's Supercharging support page has all the details as well as a tool to look up the different rates.
Tesla's recently announced Supercharger idle fees will apply to all Model S and X cars to help encourage those with fully charged cars to open up their spot for someone else who might be waiting. A per-minute fee will be levied once the car is done charging if the station was at least half full when you started charging. If the spots aren't at a premium, i.e. the station was less than half full (so pessimistic!), the fee doesn't apply, and if the fully charged car is moved from the spot within five minute of charge completion, the fee is waived. The fee structure is $0.40 per minute in the US, $0.50 in Canada, and varying fees in other countries, all of which are detailed on the Idle Fee support page.
Tesla claims the changes were not put into effect to make money but to discourage short-distance fill-ups at Superchargers and improve the charge experience for customers, something that will be more important the more Teslas are on the road. To that end, Tesla says it will invest the money collected in fees into new Supercharger stations. And while it will no longer cost nothing to get around, it'll be relatively cheap, with the company estimating the trip from LA to San Francisco (likely a popular one for Tesla owners) to cost about $12, or about $120 if you have the time and patience to make the trip from LA to NYC.
It's also important to note that the upcoming Model 3 will not include unlimited charging, but Tesla hasn't said whether it will get the same 400 kWh annually of the current cars or if it will get a different plan.