Saves can be a few different places, depending on your platform and how PPSSPP is installed. They're always in the "memory stick" which is just a folder. Inside this folder, saves are usually in PSP -> SAVEDATA.
If you extracted the zip somewhere in your Downloads or Desktop, the memory stick will be in "memstick" next to the PPSSPP program.
When PPSSPP can't create files next to the program (usually because it's in Program Files or something), it puts them inside My Documents. Look for a PPSSPP folder.
The memstick is just the same as your storage - it's meant to be just like the PSP.
The memstick is found in your "home" or "user" folder, inside ~/.config/ppsspp (you can select Go To and paste that path into the Finder or other file manager.)
If you long press on a game, you should see the "game id" at the end of the title. This might be something like ULUS12345.
Different regions have different codes - and often create different save files. If you're trying to load a save from a European game with a US game, it probably won't work.
Games almost always have a separate folder inside SAVEDATA for each save, and it starts with their game id. Often, it might be ULUS1234501 when it's the "01" save for "ULUS12345". This is the same structure you'll see on a real PSP or in downloaded save data.
Some games lock savedata to a specific device using the MAC address (which is a unique identifier in the WLAN chip).
One known affected game is Invizimals, I'm sure there are more but I can't find a list right now.
Mac addresses look something like 1c:ba:24:f3:a9:57. To load savegames affected by this, you will have to change PPSSPP's MAC address (in Network settings) to match the device the game was originally saved on. If that was another instance of PPSSPP, just write down its MAC address and put it in your local instance.
To determine the MAC address of a real PSP, go into Settings / System / System Information. Many PSPs will show their MAC address right there.
However there are some PSPs that won't. In such a case, you can run this homebrew app on your (modified) PSP to find out (it'll print the MAC address on the screen when you run it).
WARNING: Save the MAC address currently set up in PPSSPP before you modify it, in case some of your other saves are locked to it. In the future, we might make this automatic somehow.
Corrupted save data can cause crashes or glitches, and may not load. Some ways you can get corrupted data:
Note that sometimes games have bugs - some even delete your save data depending on how much free space you have.
Saves work pretty well these days, but sometimes there are bugs. Check the PPSSPP forum to see if other people have the same problem with that same game.