I want to open a google adwords account, but I'm afraid of racking up hundreds of dollars in costs and fees. How can I make sure I only spend a small amount of cash ($20 per month)? I don't want to spend a whole lot of money by accident when I'm just experimenting with the web application. I just want to test it out basically with real money.

You could implement the AdWords budget optimizer. This allows you to set a maximum monthly amount that you wish to spend and will optimize your CPC's to ensure you get the most value for money.

Aside from that, there is a general setting for Daily Budget, so you can enter $20 and once you hit that amount your ads will shut off.