The Uno has a 5V linear voltage regulator on the board, but it doesn't have a lot of heat sink area to dissipate heat. The power in watts that needs to be dissipated is given by (Vin - 5) * I, where Vin is the input voltage (e.g. 12) and I is the current used by the Uno and anything else that's connected to it, in amperes. The power supply on the Uno board can supply significantly more power than the Uno itself needs. This is very handy when you're adding things on. But everything has limitations. Asking the little regulator on the Uno to dissipate even a couple of watts will cause it to become quite warm. I tend to be conservative, and will stick a finger on the regulator, and if it's warmer than I like, then I will switch to an external power supply.
I use these wall warts a lot. They provide 9V input to the Uno. Projects that draw more than 250-300 milliamps (0.25-0.30 amperes) tend to get the regulator warmer than I like. That's just my personal preference and gut feel, nothing else. I also use these little power supplies when I need more current. It may be a good idea to add a heat sink to the regulator, depending on the current drawn and the input voltage.
Hope this helps.