This is very simple. You must get the "Grown to Maturity" award for 25 plant species. You can check your progress in the Journal, under Player Awards, then Plant Awards. When you get an award, the pop-up message will say "Award for growing _________".
To grow a plant to maturity, simply make sure it has enough water that it doesn't die. To make it easy, get the One Pour Wonder can, available from Ivor Bargain at level 24. Also keep an eye out for visitors that eat the plants in question, In particular, you may want to have two Tafflies resident to prevent any others from visiting. I'm not sure if they can be eaten before maturing, but it may be possible that you don't get the award if a plant is eaten right away. I've had something similar happen with variant awards before.
You do NOT need to fertilize the plants to get this achievement. Doing so will give you the "Catalyzed x3" a.k.a. "Bonus Growth" award if you do it correctly, which counts towards the Master Horticulturist achievement. However, if you're going out of your way to plant new species just to get this achievement, you might as well fertilize them since it'll count towards the other one as well. For specific instructions for fertilizing each plant, you should look at the plant's page on the Pinata Island wiki:http://pinataisland.info/viva/List_of_Viva_Pi%C3%B1ata:_Clas...
or see my guide for that achievement.
If you do not have access to 25 different types of plants yet, talk to Seedos as much as possible and hope he drops something you don't have. It will then become available in the store. You might also gain new seeds by leveling up, as I don't recall Seedos ever dropping e.g. Bird of Paradise seeds but I could buy them after a certain point.