Hopefully most mappings work right out of the box, but if you run into issues follow these guidelines:

Are you getting responses from the hardware? Verify by testing.
If yes, go to step #2. If no, try this.
If your controllers are responding, next step is to determine what type of game you are having problems with:

Steam Game

If the game that you are having issues with is a Steam game (you bought it on Steam):

First, we have a list of recommended Steam titles. These games have been tested and work with the Polycade controls. If you want to avoid issues, stick to this list. Note: Our list is not exhaustive, and there's tons more games on Steam that will work, we just haven't tested.
Launch the game, and attempt to find a way to remap the controls in the in-game settings. This is different for every game, so will be case-by-case.
If you can't find a way to remap the controls, it's possible this game cannot be made to work. Steam allows no-questions-asked refunds as long as you have played the game less than 2 hours.
If you are super techie, you can try using a software like XPadder to remap the controls, but in our experience this is hit or miss and will almost certainly cause a bunch of undesired side effects.

Arcade ROM

To remap controls:

Launch the game and then press TAB on your keyboard.
We'd recommend sticking to the "Input (This Machine)", as this will only modify the settings for the current game. Avoid using "Input (general)" as this will change the settings for ALL arcade games, which could have unintended consequences.
Navigate using the arrow keys on your keyboard. Press Enter to change a setting, then press the key/button you want to map it to. If you mess up, press Delete on your keyboard to clear a setting.
When you're done remapping, press TAB to close the settings and then press ESC to save your settings.
WARNING: The current version of Polycade AGS will overwrite your custom configs the next time it updates. We're working on fixing this to allow for customization in a future update. If you plan to edit your configs, we'd recommend making backups of your custom configs.

Console ROM

Most consoles use RetroArch for the emulator, so the mapping instructions are the same for NES, Genesis, SNES, etc.

(copied from Reddit): Load a game and then press F1 to enter the quick menu. From there you can go to Controls and change your User 1 buttons around with left and right. Then press confirm (enter on the keyboard) on the Save Core Remap entry to save those just for the console core (i.e. NES) you have loaded.

To change all at once, double click the retroarch.exe located here:
C:\Users[YOU]\AppData\Roaming\polycade\emulators\retroarch\
Go to Settings -> Input -> Port 1 Binds (for player 1, Port 2 Binds for player 2, etc)

Game from the Polycade AGS store
All of these should work right out of the box. Please let us know if one doesn't. We are currently working on fixes for asteroids & black widow. Otherwise, if you have an issue with an AGS game that is a retro arcade or console title, you can use the instructions above for "Arcade ROM" and "Console ROM" to remap the controls.