Shareware
Shareware, sometimes called a demo or trial, is a software distribution model in which a program is distributed for free with the hope that users will appreciate it enough to send money to the creator. The term came into use in the early 1980s and a handful of developers found success with it. Occasionally developers would distribute the their software with nag messages asking the user to pay money to "register" the software, which would eliminate the nag message, making it nagware. Some developers would release a product as shareware, but required the user to pay to unlock additional features, making it crippleware.
In the early 1990s Apogee Software began distributing games for free, but offered additional premium episodes for those players willing to pay for them. This business model, later called the "Apogee model" was quite successful for the company and quickly dominated the shareware market and companies like Epic Megagames quickly adopted it. By allowing players to get sucked into their games, they were far more likely to want to buy the game than from only seeing a box in a store. Despite it's success, many "pure" shareware developers didn't like it because they felt like the software wasn't truly given out for free, but was more like a demo. In fact, the Apogee model even pervaded commercial software as it became extremely common for them to release demos or trials of the early portions of their games. Now that cheap online video makes advertising so common, demo versions of software aren't nearly as popular.
Personal
I respect the original shareware model of only paying for software that you find useful, however, I'll admit that I've only ever paid for a tiny amount of the free software I've used. I have less respect for the Apogee model, but I recognize how effective it is. I prefer either model over those programs which call themselves free, but are actually crippleware, nagware, or adware in disguise.