Yes, your system can support a 2.5" SSD replacing your current HD. Make sure you get a large enough drive. You'll want to leave no less than 1/3 of the drive free if you have anything smaller than 512 GB SSD anything larger you'll need 1/4 free. Unlike a HD, SSD's need more free space.
Here is the IFIXIT guide you'll need to follow: MacBook Pro 13" Unibody Mid 2012 Hard Drive Replacement
But before you physically replace the drive you'll want to prep your new SSD drive. You'll want to get a SATA to USB adapter like this one: Startech 2.5" SATA to USB adapter.
To prep your new SSD drive connect it via the adapter using Disk Utility from your system format the drive with a GUID partition map and Mac OS Extended (Journaled) petition. Download the OS installer to your system and run it if you don't already have it. At the end the installer will ask you if you want to migrate your user, apps & data to your new drive. Select as you need the user data and migrate your stuff over.
Once the system restarts you're ready to swap your drives. One word of caution here. This systems SATA drive cable is fragile so do be gentle. If you encounter any issues I would recommend replacing the cable as the first thing. Here's the IFIXIT guide to do that: MacBook Pro 13" Unibody Mid 2012 Hard Drive Cable Replacement.