Google originally introduced Google Checkout as a method for online payments. At the time it was introduced, 2006, Google wasn't a major vendor of anything but keyword advertising. There was no Android Market/Google Play Store, but there was PayPal, and the move was widely seen as an attempt to compete with the popular payment system. (PayPal has even sued Google over the accusation that Google Wallet uses trade secrets provided by former PayPal employees later hired by Google.) Google Checkout was intended to provide online merchants and easier way to accept credit card payments without having to pay huge transaction fees.
Over time, Google did roll out the Android Market (which is now the Google Play store) and start selling Android apps, music, books, movie rentals, and even devices. Google needed a way for users to pay for their purchases, and phone companies were initially reluctant to allow Android Market purchases to be passed through to the user's phone bill. Google Checkout filled in the gap.
Meanwhile, Google Wallet was designed as a mobile payment system that used your phonee as if it were a credit card. Google Wallet allows NFC-capable phones to "tap to pay" at participating swipe stations. Initially Sprint was the only network to support this method, and the offer was restricted to a limited number of credit cards. You otherwise had to make a pre-paid credit card with your Google Wallet account and then use it to pay your bill instead of just pulling out the credit card you were probably still carrying around.
Google Wallet has now merged with Google Checkout, so all your Google Play purchases happen with Google Wallet. In addition, Google Wallet now supports all major credit cards, so you really could just carry around your phone as if it were your credit card. So in addition to being a mobile payment system, it's a cloud-based payment system. You don't need to have a phone to use the cloud-payment features.
Carrying around your credit card on your phone is a potential security threat. What if someone steals your phone? Do they now have access to your credit card numbers? What if they break into your Google Account? Can they steal your credit card numbers that way? In order to use Google Wallet as a mobile payment system, you also have to use a 4 digit PIN. That means you're not using your regular Google account password. If you know someone has stolen your phone, you can disable access to Google Wallet remotely.
You should also set up two-step verification of your Google accounts, just to make it a little harder to hack. You should do that anyway.