What is Auto-thrust?
In its basic form, it is just like the cruise control in a car.*

Here is a photo of the autopilot control panel on a Boeing 777. Note the "IAS 200" window on the left:



IAS stands for "indicated airspeed". When engaged, this setting tells the auto-thrust to adjust engine power to maintain 200 knots of airspeed.

Just like a car, when you go uphill you need more gas to maintain a constant speed, and when you go downhill you give it less gas. If the downhill is steep enough, you may have to use some brakes to maintain a constant speed.

Same thing happens when a plane pitches up to climb or pitches down to descend. Airspeed, vertical speed and engine power are closely related - changing any one will impact the remaining two. This is the fundamental of how planes fly, and likely your first ground school lesson if you learn to become a pilot. Auto-thrust is a component of the autopilot system which manages engine power. Note that I used constant airspeed as an example in the previous paragraph: advanced autopilots have many modes, and it can be set to maintain constant vertical speed instead of airspeed. But in all cases, auto-thrust manages engine power.

* Well said TomMcW