How do you transfer to a new laptop from an Internal Hard Drive removed from an old laptop?
Solution 1:
The term you are looking to search on would be "SATA USB enclosure." Typically it will be a box that you slide the hard drive in. You then attach the USB cable to a PC, and it should come with an external power supply if needed. Laptop hard drives are 2.5" so make sure you look for a 2.5 SATA USB enclosure.
I think many of the 2.5 enclosures don't even need external power supplies.
When you put the hard drive in there, power on the enclosure, and attach it to the computer, it should pop up as another drive in "My Computer" on Windows, or simply appear as another drive on Mac or Linux.
Once you copy your data, you may want to simply keep it around as another travel drive, for backup use or similar.
Solution 2:
You don't actually need an enclosure. A "USB to SATA/IDE cable", as you mentioned, and which you should be able to find for cheaper, will also work.
...and if your laptop is fairly new, you can make do with an even simpler USB to SATA cable, as described here: http://www.pcworld.com/article/2036718/why-its-a-good-idea-to-own-a-usb-to-sata-adapter.html