TrueType was invented by Apple as a competition to Adobe's PostScript Type1. Both TrueType and PostScript fonts became the standard file formats for fonts for the past 3 decades or so of desktop publishing. In terms of your average designer, the differences between the two are relatively unimportant.
OpenType was designed to replace these and was created initially by Adobe and Microsoft. It's basically a newer format that is more robust. For a designer, the primary benefits of OpenType over previous formats are a) an much larger character set and b) automatic alternative character and ligature support (for software that supports it). Wikipedia covers it well: