Last month the household surveys suggested 651,000 new jobs yet the official number was 199,000, a difference of 452,000. For the year, the household surveys suggest 356,000 more jobs than the official number.
Last year, we had a record number of new businesses start up. Significantly more. In 2021, 53% more businesses were started than in 2019. Those new businesses don't get surveyed because they're not yet on the list for BLS. [Per Austan Goolsbee]
On top of that, people who start unincorporated businesses aren't counted as having new jobs. They're not employees. So if you leave a job to start a business, you will actually show up as one less employee in the job count. [Per Jason Furman.]
One more item of note. When BLS reports 500,000 new jobs, that is a net number. Every month people quit, retire, get laid off or fired. So a month in which you might report 537,000 new jobs (the average for 2021), you may actually count about 4.5 million new jobs and 4.0 million jobs ended, for a net of 500,000.
Quits were at a record level last year. Compared to the rest of this century, quits are up by about 1.3 million. Per month.
So the economy is creating a record number of net new jobs in spite of the fact that 43 million people quit their job last year. (And that is just through November.)
Last year, the economy created and brought back a total of 6,448,000 jobs. Again, that is net. That shatters the old record (set in 1946, the year after World War 2 ended) by 2.2 million. And it may well prove to be an undercount because the rate of new business formation is up by 50%.
The phrase you are looking for is "strong job market."