The SIM identifies your account to the your cellular carrier. If you move the SIM to another compatible and unlocked device, you transfer your service and your phone number to that new device.
SIM cards have always been tied to a given carrier. If you want to change the carrier you connect to, you need to get a new SIM.
One surprise feature in today's launch of the new Apple iPad Air 2 and iPad Mini 3 was that the cellular models will no longer require you to pick a carrier at purchase time to get the proper SIM bundled with your new device.
Instead - every new cellular iPad will come with a magical Apple SIM - the first SIM to ever support multiple carriers at once.
With the Apple SIM, right from within the iPad's settings app you will be able to pick a carrier and sign up for short term pay-as-you-go cellular data plans.
If you travel to someplace where another carrier has better speeds or signal, with just a few clicks you can change which network you are connected to.
Initially AT&T, Sprint, T-Mobile, and UK carrier EE will be supported - with many more promised in the future. If you want to use a non-supported carrier (like Verizon), the iPad's are all completely unlocked - you just need to go back to the old way of physically swapping the SIM card.
The flexibility of the Apple SIM potentially makes international travel a lot easier, and even in the US it turns the iPad into a great backup data connection provider. If you are ever someplace where you need to get online and your main provider isn't cutting it, perhaps a week's worth of data on another carrier might get you through.
This also opens the door to price competition - with carriers in a given area having an incentive to offer especially attractive tablet data deals.
I am looking forward to hearing how well the Apple SIM works in practice - but in general this seems like a potentially great feature.
UPDATE (10/24/2014): Now that the new iPads are shipping, people have discovered that AT&T is not fully supporting the Apple SIM after all. Once you select AT&T, the SIM becomes locked - and you can not change to another carrier without getting a different SIM. Read our news story for details.