My ISP has just installed a new modem with an ethernet connection, which I have plugged into my Airport Express. I'm able to connect to an open network in the neighborhood but not to my own network that I had set up before. I have done some troubleshooting, the modem is working when connected to my computer. It may just be the password, it's not accepting the password that I thought I had used to set up the Airport Express when I first bought it. Should I be using the same password as in my Accounts on my Mac's System Preferences?
... Would the network password be the same as my Mac's password?
It can be if you configured it that way, but since that is apparently not working for you it's probably not the same. You can reconfigure the Express easily enough using AirPort Utility - it is in your Utilities folder, which is found within your Applications folder.
Launch AirPort utility, then reset the Express:
To reset your Express:
Press and hold the reset button with a pen or pencil for 1 full second. The light (LED) will flash amber, indicating that the AirPort Express is in soft reset mode.
This will enable you to use AirPort Utility to reset your network password.
Note: If you do not make your changes within five minutes of pressing the reset button, AirPort Express will return to its previous configuration. The light will change back to solid green (or whatever state it was in before the reset).