Once upon a time when I was younger, I used something called the "wallet." The wallet was a small bag holding paper and metal. You used the metal and the paper to pay for the products you wanted to purchase. The problem with the wallet was that you were always too lazy to pay with the metal - you always used the paper. And when the value of the paper wasn't the same as the value of the products, you got metal back. The amount of metal in the wallet soon added up, and the wallet became really heavy. One other strange thing was that you had to visit a machine to get more paper for your wallet from your bank account. Sometimes these machines were closed, and you had to run around the city to find another machine. When you got robbed by a thief, the thief could use your paper and metal the same way the thief used his/her own paper and metal. But that was a looong time ago because a few companies have begun to replace your wallet with your phone.
Google Wallet is like a virtual wallet on your Android smart phone. The application is password protected, and if you get robbed by a thief, you can disable the account from any other computer. It is possible to use the product when you are shopping online as well as offline in a physical store. When using Google Wallet offline, you need to make sure the store/taxi/bus has an NFC terminal. The main drawbacks today are that you can only use the product on Android in the US, and you need an NFC terminal. The NFC terminals are spreading around the world, but they are far from available everywhere, according to this list: List of applications of NFC. On the positive side, Google Wallet is free and you don't have to pay a fee to use it. God knows how Google plans to make money from it - but Google is also famous for making free products. Their goal is always to first find users, and then find out how to make money from the users.
Square is a US startup founded in 2010. To be able to use Square to read a credit card, you need a square that you can attach to your iPhone, iPod Touch, iPad, or Android device. You now need to install the app, and you can now receive payments from people with credit cards. If you recall from the section about Google Wallet and that Google Wallet needed an NFC terminal to be able to work, Square only needs the little square to be able to work. Square is not free of charge, you need to pay 2.75 percent on every credit card transaction. Business seems to be fine, Starbucks announced this week, that their customers now can use Square to pay for their coffee at 7000 stores. It is also possible to use Square to pay for products, using only your name. You connect to the store through your phone, the store will now receive you name and a photo of yourself, and you just need to say your name when you purchase the product. The cashier will now look at your photo and register the purchase. The service is currently limited to the US market.
iZettle is a Swedish startup founded in 2010. It has the same business model as Square above, and I'm not sure who got inspired by who. Square and iZettle are not competitors today, Square is only available in the US, while iZettle is only available in the Northern parts of Europe. Both are planning to expand, so they are probably going to compete with each other in the near future. iZettle is not free of charge, you need to pay somewhere between 2.75 and 3.75 percent on every credit card transaction.
DigiMo is an Israeli startup and is similar to Google Wallet, but you don't need the NFC Terminal. They have replaced the NFC Terminal with a QR-code at the cash register. Scan the QR-code with your phone and you are done. It is also possible to transfer money from a nearby phone. I'm note sure why Google Wallet is not using the same system - maybe the QR-codes are less safe?
boxPAY is an Irish startup founded in 2011. The company's business model is based on the SMS-technology. You need your phone number and a pin-code to complete the purchase which is automatically added to your phone bill.
Venmo is a US startup founded in 2009. It is possible to use Venmo by using their iPhone/Android app, or by using SMS-technology. It is easy to pay someone, just open the app, choose the person you would like to pay money to, write a description, and press the pay-button. You can pay someone who is not using Venmo, but that person has to register and become a Venmo-user to be able to transfer the money to his own bank account.
Update! I've written a second part: These companies are again trying to replace your wallet