(1) Startup (or start-up, aka System startup) refers to the short period of time, or system state a computer is in, immediately after switching it on. This is also called the boot-up period.
(2) The word start-up (or startup) is also synonymous with startup company. A startup company is a company recently formed, usually till IPO.
This latter meaning rose to popularity in the late 1990s during the dotcom boom, where much stock market speculation and hype surrounded small hi-tech start-up companies seeking early IPO and promising enormous future profits. Many such start-ups started as spin-offs from university research groups.