When you switch into what different browsers refer to as private or incognito mode the browser changes its behavior. For example it won't save history, and any cookies will be deleted as soon as you close the tab or window (if they are ever saved at all). This has the effect of hiding the online activity in that browser.
But these changes in behaviour are local to that browser/machine. They don't affect what happens everywhere else in the chain. Search engines, ISPs, etc. don't care about what you are trying to hide.
For example – the data that flows between your browser (regardless of mode) and a website goes through your ISP and other networks. They will still be logging your activity, and if for example you were illegally downloading files incognito mode will not help you. This is important if you are running a business and let employees use your internet connection – if they share files or engage in other illegal activity online it can still be traced back to you (not them, but you as the owner of the connection). Don't let employees tell you that incognito mode is protecting you as the business owner.
And if you are an employee you should assume that the network administrator can see everything you are doing regardless of incognito or private mode. Again – those options just affect your browser, not the employers networking hardware. It is still very likely to be logging all of your online activity.
It's worth noting that some small local businesses, like flower shops, will sometimes use incognito mode to better test their search performance. This is likely to give a better, or at least more neutral result (less biased by past searching and clicking history) but is still not
The intent of private/incognito modes (hiding online activity) can also be undone by user behavior. If for example you download a file while in incognito mode it will remain on your machine, not simply disappear when you close the tab. Same thing with bookmarks – any that you create during a private session will remain. The various safe modes are designed to protect you from inadvertent tracking (your activity won't show up in the browser history) but if you choose to create book marks or download files they can't help you.
And of course as soon as you log in to a website that site will be tracking you. If for example you log in to check your gmail and then do some searches on Google then Google will still be tracking and saving those searches on their end. They will still be hidden from your local browsing history, but they will absolutely be tracked on other end. If for example you were logged in to your LinkedIn account and visit other profiles those visits will be logged and visible in LinkedIn, which does not respect your browser privacy settings.
There is more good information on the different modes and what they mean at Gizmodo.