What is this thing next to my sink?
If the top item is near your kitchen sink then this probably an air gap for your dish washer drain system. The purpose of the air gap is to prevent back flow siphoning from the sink drain back into the dish washer. 



The lower item is definitely a timer unit. The dark colored thing under the timer dial appears to be a switch that has positions of OFF, FULL ON and TIMER MODE. You should be able to use this switch to determine just what things this timer turns on and off. A common use for a timer like this is to turn certain lights in your dwelling on and off according to a fixed schedule. If it connects to internal lights then the purpose it to make it look like people are home whilst you are away on vacation. It may also be tied into exterior lights such as an entry way light, lights on the front of your garage and/or sidewalk lights. In this case the purpose would be to automatically turn these lights on of a few hours in the early evening and then turn them off automatically when you have gone to bed. 

More commonly it is used to control heating or other high-load electricity/gas-powered systems as many tariffs offer cheaper power overnight.