"Start-up companies" are generally best described in the simplest terms - companies that have been in existence for less than 12 months. Start-up jobs are the jobs that those companies are looking to fill.
Here is an interesting infographic all about "start-up jobs" in the USA. Please note that size of the call-out boxes in the graphic where the jobs within states are being show. Visually, the size of the state call-out boxes and text seems to imply
how big the numbers are for each state; however, that’s not the case and
it’s misleading to the reader. Massachusetts, for example, had a higher number of start-up job
posts than Texas, but the call-out text is much large for TX versus MA. That being noted, it's still an interesting look at where new companies are creating new jobs.