New SUPEE-10415 patch can be downloaded as usual from Downloads page:
https://magento.com/tech-resources/download#download2133 or installed as a regular Magento upgrade via Downloader (it is included in Magento 188.8.131.52 version).
You can install it in the same way as previous patches or by upgrading to Magento 184.108.40.206.
To apply the patch you need SSH access (shell access actually, SSH is just most used way to get shell access) to the server. If you have no SSH access, you can refer to How to apply SUPEE-10415 without SSH.
If you wish to save time and have us to install these patches for you, simply click here to order installation.
Step 0: Preparations
Note: Make sure to Disable Magento Compiler at System > Configuration > Tools > Magento Compiler and clear compiled cache.
Step 1: Verify your Magento version
As you can see in the example, it is Magento 220.127.116.11
Step 2: Download corresponding patch
The patch should be downloaded from https://magento.com/tech-resources/download#download2133
Make sure to get the right file corresponding to your Magento version.
Step 3: Place patches into Magento Root directory
Upload your files into Magento root directory.It is important to place patch files directly into Magento root directory and execute it also directly in Magento root directory.
Step 4: Run the patches
Step 5: Verification and flush of Magento PHP opcode cache
Flush Magento caches: Navigate in Magento backend to System > Cache Management and flush Magento cache and CSS/JS caches.
If you use PHP opcode caches (OPCache/APC/XCache/eAccelerator) make sure to flush it after patching (or restart webserver), otherwise code will continue to run from caches.
Test that your store is working. Test Checkout process.
If you have any difficulties with applying the patches please let us know in comments, so we can find the solution together.