The existing answer has most of the properties that make a sensor good, but I would add one more property:
Today, with mirrorless cameras, the camera may require autofocus features from the imaging sensor. Without special autofocus support, only slow contrast detect autofocus (CDAF) is possible. With special autofocus support, one can use phase detect autofocus (PDAF). For example, the autofocus support could be Canon's dual pixel autofocus which is a form of PDAF.
The autofocus features may be important too with DSLRs when shooting using the live view mode. Then, the only autofocus features available are CDAF, flip mirror + do PDAF (sometimes said to be quick mode as it's quicker to flip the mirror than to do CDAF), and only if the imaging sensor has it, on-sensor PDAF. Generally, on-sensor PDAF is so much better than CDAF that it may not be possible to even select CDAF if on-sensor PDAF is supported. Furthermore, when shooting video (out of the scope of this site but necessary to mention), the autofocus features are needed.
You really don't want to use CDAF due to its slow speed.
On-imaging-sensor autofocus is today better than using a dedicated focus sensor. For example, you get more autofocus points with the on-imaging-sensor autofocus than you do with a dedicated focus sensor. Furthermore, you get autofocus even using f/11 lenses (dedicated sensors do only f/5.6 or f/8), and you also get autofocus down to -5 or -6 EV if using f/1.2 lens (for example, 5D Mark IV does autofocus only down to -3 EV).
Most importantly, what do these things look like in actual real-world images, and how can I recognize them by looking it photographs rather than just review charts and graphs?
If you have a photograph that is out of focus, you will immediately recognize it. Nothing is sharp on the subject. Actually, focus problems are something you won't see in review charts and graphs. Only using the equipment in real-life conditions will tell you how well autofocusing works. With on-imaging-sensor AF, it works just fine.