You press a button on your controller and, a moment later, your character jumps. That "moment" is input lag: the time between your press and the change you see on the screen. Most of it is not your console's fault. A big part is decided by your TV, and the fix is usually one switch.
Input lag is the total delay from pressing a button to seeing the result. It is measured in milliseconds (ms), where 1,000 ms is one second. A game running at 60 frames per second draws a new picture every 16.7 ms, so you can think of 17 ms as "one frame". At 120 frames per second a frame lasts 8.3 ms.
The delay is a chain. The controller sends your press to the console, the console works out the next frame, the TV receives it, the TV processes it, and finally the panel lights up. The console and controller part is small and the same whatever TV you own. The TV processing step is the one that varies a lot.
TVs are built to make films and sport look good. To do that they sharpen edges, remove noise, smooth motion by inventing extra frames and tweak colours. Each of those steps needs the TV to look at several frames before it shows one, and that waiting costs time. For a film nobody notices. For a game, you press a button and the picture arrives late. With heavy processing switched on, many TVs add dozens of milliseconds.
Almost every modern TV has a Game Mode. It switches off most of the picture processing so the image goes to the screen as fast as possible. The picture can look a little plainer, but the lag drops a lot. Good TVs in Game Mode can get under about 20 ms, and some reach under 10 ms.
Newer TVs with HDMI 2.1 can do this for you. It is called ALLM, short for Auto Low Latency Mode. When the console tells the TV "this is a game", the TV switches to its low-lag mode by itself, and switches back when you watch a film. If your TV has it, check that it is turned on in the TV's settings, because some TVs keep it off by default.
120 Hz means the screen can show up to 120 pictures a second. Games that offer a 120 fps mode feel smoother and each frame is shorter, which trims lag a little more. You need a TV that supports 4K at 120 Hz, a cable that can carry it and a game that has the option.
VRR (variable refresh rate) lets the TV follow the console when frame rate wobbles, which removes tearing and stutter without the delay that older fixes like V-Sync can add. It is another HDMI 2.1 feature.
One warning: Game Mode cuts lag but does not make a slow panel respond faster. Blur or smearing behind moving objects is a different problem and comes from the panel, not from the processing.
If you only play slow adventure games, input lag matters less and you can pick a TV for picture quality. If you play shooters, fighters, racing or rhythm games, put low lag first. Here is what to check on the box or the specification page:
The quickest win costs nothing: open your TV's picture settings, switch to Game Mode, and play your favourite game for ten minutes. Most people notice the difference at once.