You may have noticed that the real deal Swype did not come pre-installed on your Galaxy S3. Many S3 users have encountered a parsing error that reads, "Parse error" "There is a problem parsing the package.". This is a bit of a pain in the arse and I have no doubt it'll be rectified shortly. However, in the meantime you have two options...
Use the built in swype-style keyboard. To activate it go to Settings > Language and input, and then tap on the gear next to Samsung keyboard. THEN tick the box next to Continuous input. You should now be able to "swype", although it doesn't appear to work on all text fields and it doesn't have the Swype shortcuts for Select All, Copy or Paste.
If you want the real Swype you can download the installation file here thanks to one of the kind members on Swype Forums.
IMPORTANT: When you first install it, Swype may appear really narrow on the left hand side of the screen. Rotate your Galaxy S3 to landscape mode. Swype should take up the full width of your Galaxy S3 screen and if you rotate it back to portrait it should correct itself.
Thanks to Patchseven on XDA-Developers, here is another link to a Swype installation file which appears to be specifically for the Samsung Galaxy S3. It worked a treat for me, and fixes the small keyboard problem found in Option 2.
UPDATE AGAIN: Sype has just released a new beta version of their keyboard and I can confirm that installation on my Galaxy S3 completed without a hitch. I didn't get any of those Parse Errors this time. If you've previously been using one of the above alternatives, you'll need to uninstall the existing version of Swype from your S3 first. Then get yourself over the Swype.com to download the new installer.