Every once in a blue moon, you'll run into a problem with your Drupal site where the solution involves patching core. Start by making sure that you have a safe place to apply your patch. This would be a 'dev' copy of the website, either running locally on your own computer, or in a development environment on the same server (or one that's identical) to your live site. *Never* apply patches directly to your live website - *especially* core patches!!!

1) Start from the root of your Drupal site.

[.../drupal-7] 

3) Download the patch file from drupal.org

[.../drupal-7]  wget http://drupal.org/files/606598-core7-port-imagestyles-175.patch

If you don't have wget on your computer (with a mac you will need to install Xcode) then you can do the same thing with curl, as follows

[.../drupal-7]  curl -O http://drupal.org/files/606598-core7-port-imagestyles-175.patch

4) apply the patch

[.../drupal-7]  patch -p1 < 606598-core7-port-imagestyles-175.patch

If everything goes well, you'll see the files that are patched:

[.../drupal-7/sites/all/modules/contrib/panels]  patching file modules/image/image.admin.inc
[.../drupal-7]  patching file modules/image/image.api.php
[.../drupal-7]  patching file modules/image/image.install
[.../drupal-7]  Hunk #2 succeeded at 454 (offset -1 lines).
[.../drupal-7]  patching file modules/image/image.module
[.../drupal-7]  Hunk #1 succeeded at 347 (offset 5 lines).
... etc
[.../drupal-7]  patching file modules/image/image.test
[.../drupal-7]  Hunk #2 succeeded at 462 (offset 56 lines).
... etc

5) Clear your caches, and go test your new fix/feature!