You don't have to buy anything (though they certainly won't turn down some money to fund the work they've been doing), and you don't have any kind of licensing or contract with Canonical.
If you're computer is connected to the network, all you have to do is open up System Updates. It should tell you that there's a new version of Ubuntu available. Just click the button that says "Upgrade" and it will do the rest for you.
If you don't see an option to upgrade, then you'll need to go into "Software Sources" and on the "Updates" tab, make sure "Notify me of a new Ubuntu version" is set to "For any new version." Then, when you have System Updates search for new updates, it will tell you that 12.10 is available and you can upgrade.
The difference between 12.04 and 12.10 is that 12.04 is a "Long Term Support" (LTS) release. This means that Canonical will offer support and updates for a longer time frame than for the other releases. Additionally, you can upgrade straight from LTS release to LTS release, so you can go from 10.04 to 12.04 (and, when it comes out, to 14.04) without having to upgrade through every release in between.
Generally speaking, it's recommended to stay current on Ubuntu versions, so that you have the latest versions of software (many software vendors won't support older versions of software and will tell you to update to the latest if you have problems), especially as the time for a new LTS release approaches. However, sticking to an LTS version usually has the advantage of stability. The packages included in an LTS release are usually tested longer and can be more stable.
Additionally, whether to upgrade also depends on your hardware. Is it a new machine, or are you just looking to get a couple more years out of a decade-old workhorse? If the former, then upgrading not only won't hurt, but may actually provide improved support for your hardware. If the latter, then you may be better off sticking to 12.04 to prevent issues with the hardware not being able to support the demands of the software (both for 12.10 and future releases).