NOTE: JS's keypress is deprecated:

https://developer.mozilla.org/en-US/docs/Web/API/Element/keypress_event

The jQuery solutions in this answer may still be relevant.

KEYPRESS (enter key)
Click inside the snippet and press Enter key.

Vanilla

document.addEventListener("keypress", function(event) {
  if (event.keyCode == 13) {
    alert('hi.');
  }
});

Vanilla shorthand (Arrow Function, ES6)

this.addEventListener('keypress', event => {
  if (event.keyCode == 13) {
    alert('hi.')
  }
})

jQuery

$(this).on('keypress', function(event) {
  if (event.keyCode == 13) {
    alert('hi.')
  }
})

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>

jQuery classic

$(this).keypress(function(event) {
  if (event.keyCode == 13) {
    alert('hi.')
  }
})

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>

jQuery shorthand (Arrow Function, ES6)

$(this).keypress((e) => {
  if (e.keyCode == 13)
    alert('hi.')
})

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>

Even shorter (ES6, ECMAScript 2021)

$(this).keypress(e=>
  e.which==13&&alert`☺`
)

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>

Due to some requests, here is an explanation:

I rewrote this answer as things have become deprecated over time so I updated it. Just for info, it is not about "keydown", it's about "keypress". So some non-character keys like "Esc" aren't supposed to work like that but I'll explain.

I used this to focus on the window scope inside the results when document is ready and for the sake of brevity but it's not necessary.

Deprecated:
The .which and .keyCode methods are actually considered deprecated so I would recommend .code but I personally still use keyCode as the performance is much faster and only that counts for me. The jQuery classic version .keypress() is not officially deprecated as some people say but they are no more preferred like .on('keypress') as it has a lot more functionality(live state, multiple handlers, etc.). The 'keypress' event in the Vanilla version is also deprecated. People should prefer beforeinput or keydown, keyup today.

Performance:
The faster the better. This is why I prefer .keyCode even if it's considered deprecated(in most cases). It's all up to you though (Commited 2020).

Performance Test

Use event.key and modern JS!

No number codes anymore. You can use "Enter", "ArrowLeft", "r", or any key name directly, making your code far more readable.

NOTE: The old alternatives (.keyCode and .which) are Deprecated.

document.addEventListener("keypress", function onEvent(event) {
    if (event.key === "ArrowLeft") {
        // Move Left
    }
    else if (event.key === "Enter") {
        // Open Menu...
    }
});

Mozilla Docs

Supported Browsers

Don't over complicate.

  document.addEventListener('keydown', logKey);
    function logKey(e) {
      if (`${e.code}` == "ArrowRight") {
        //code here
      }
          if (`${e.code}` == "ArrowLeft") {
        //code here
      }
          if (`${e.code}` == "ArrowDown") {
        //code here
      }
          if (`${e.code}` == "ArrowUp") {
        //code here
      }
    }

There are a few ways to handle that; Vanilla JavaScript can do it quite nicely:

function code(e) {
    e = e || window.event;
    return(e.keyCode || e.which);
}
window.onload = function(){
    document.onkeypress = function(e){
        var key = code(e);
        // do something with key
    };
};

Or a more structured way of handling it:

(function(d){
    var modern = (d.addEventListener), event = function(obj, evt, fn){
        if(modern) {
            obj.addEventListener(evt, fn, false);
        } else {
            obj.attachEvent("on" + evt, fn);
        }
    }, code = function(e){
        e = e || window.event;
        return(e.keyCode || e.which);
    }, init = function(){
        event(d, "keypress", function(e){
            var key = code(e);
            // do stuff with key here
        });
    };
    if(modern) {
        d.addEventListener("DOMContentLoaded", init, false);
    } else {
        d.attachEvent("onreadystatechange", function(){
            if(d.readyState === "complete") {
                init();
            }
        });
    }
})(document);

Catching the event on keydown is very stable,I have used the code before no several occasion with positive result.

This link will show you keycodes, as you hit the key. Find keycode

$("#your_input_id").keydown(function(event) {
if(event.keyCode == 189) {
    alert('Dexter');
}
});