PayPal was one of the first services to let people exchange money online. It was popularized--and later purchased--by eBay, and is one of the most widely recognized payment systems on the Internet. It's used in online auctions and stores as a way to control and manage payments.
PayPal has survived throughout the years by charging fees to its sellers. Buyers in online auctions and those making donations via the service aren't charged these fees, which are generally based on the amount of the transaction.
One of the reasons people choose to use PayPal over other services is its buyer protection and anonymization services, which offer some protection against online fraud. More recently, alternate services such as Google's Checkout have garnered some of the attention in this space for similar efforts with lower fees, but PayPal should still be a top choice for your online merchant requirements, and it's now required for pretty much every eBay transactions.