As part of the conceptual and preliminary design processes of the Thirty Meter Telescope (TMT), the TMT site-testing team has spent the last five years measuring the atmospheric properties of five candidate mountains in North and South America with an unprecedented array of instrumentation. The site-testing period was preceded by several years of analyses selecting the five candidates: Cerros Tolar, Armazones and Tolonchar in northern Chile; San Pedro Mártir in Baja California, Mexico; and the 13 North (13N) site on Mauna Kea, Hawaii. Site testing was concluded by the selection of two remaining sites for further consideration, Armazones and Mauna Kea 13N. It showed that all five candidates are excellent sites for an extremely large astronomical observatory and that none of the sites stands out as the obvious and only logical choice based on its combined properties. This is the first article in a series discussing the TMT site-testing project.