Mandrake's (500) and Mandrake Seeds (800) can be purchased from the
Dogweed & Deathcap in Hogsmeade. It's far cheaper to grow your own so I suggest buying the seeds. Mandrakes will grow in any sized pot and can be made early at the potting table in the Greenhouse.
Once you have some Mandrake's in your possession equip them by holding
to open the equipment wheel. Once highlighted tapping
will pull out a Mandrake to use in combat. This will stun all enemies around you, and as the achievement suggests you need to stun 10 different named enemies. There's two methods you can do this:
Method 1:
Gather 1x Mandrake and manual save before engaging a group of enemies. Use the Mandrake to stun the group. Then reload the save, fly to another group and repeat. Continually using the 1x Mandrake until you've stunned 10 different types.
Method 2:
Grow around 5x Mandrake's and take them into a Dark Arts Battle Arena. As each wave starts use the Mandrake to stun the 5 or more enemies each round. You should get the achievement on the 5th wave if the groups have been diverse enough.