Note: both versions of rgbToHex expect integer values for r, g and b, so you'll need to do your own rounding if you have non-integer values.

The following will do to the RGB to hex conversion and add any required zero padding:

function componentToHex(c) {
  var hex = c.toString(16);
  return hex.length == 1 ? "0" + hex : hex;
}

function rgbToHex(r, g, b) {
  return "#" + componentToHex(r) + componentToHex(g) + componentToHex(b);
}

alert(rgbToHex(0, 51, 255)); // #0033ff

Converting the other way:

function hexToRgb(hex) {
  var result = /^#?([a-f\d]{2})([a-f\d]{2})([a-f\d]{2})$/i.exec(hex);
  return result ? {
    r: parseInt(result[1], 16),
    g: parseInt(result[2], 16),
    b: parseInt(result[3], 16)
  } : null;
}

alert(hexToRgb("#0033ff").g); // "51";

Finally, an alternative version of rgbToHex(), as discussed in @casablanca's answer and suggested in the comments by @cwolves:

function rgbToHex(r, g, b) {
  return "#" + (1 << 24 | r << 16 | g << 8 | b).toString(16).slice(1);
}

alert(rgbToHex(0, 51, 255)); // #0033ff

Update 3 December 2012

Here's a version of hexToRgb() that also parses a shorthand hex triplet such as "#03F":

function hexToRgb(hex) {
  // Expand shorthand form (e.g. "03F") to full form (e.g. "0033FF")
  var shorthandRegex = /^#?([a-f\d])([a-f\d])([a-f\d])$/i;
  hex = hex.replace(shorthandRegex, function(m, r, g, b) {
    return r + r + g + g + b + b;
  });

  var result = /^#?([a-f\d]{2})([a-f\d]{2})([a-f\d]{2})$/i.exec(hex);
  return result ? {
    r: parseInt(result[1], 16),
    g: parseInt(result[2], 16),
    b: parseInt(result[3], 16)
  } : null;
}

alert(hexToRgb("#0033ff").g); // "51";
alert(hexToRgb("#03f").g); // "51";

1. Hex codes use base 16 (hexadecimal), RGB uses base 10 (decimal)
2. Hex codes can be split into 3 bytes which describe red RR, green GG then blue BB, i.e. #RRGGBB
3. The # for hex isn't always written, or sometimes it may use 0x or &h etc instead (you may need to consider this depending on your envionment)

Putting these together and using a RegExp to extract the interesting bits and parseInt to convert from the String representation of the base 16 numbers

function hex_to_RGB(hex) {
    var m = hex.match(/^#?([\da-f]{2})([\da-f]{2})([\da-f]{2})$/i);
    return {
        r: parseInt(m[1], 16),
        g: parseInt(m[2], 16),
        b: parseInt(m[3], 16)
    };
}
// ex.
hex_to_RGB('#FF0110') // {r: 255, g: 1, b: 16}

When converting in the opposite direction you would use num.toString(16) and don't forget to pad the bytes with a '0' if they're smaller than (dec) 16

If you're starting with an Integer i which represents your hex colour, you can access the RGB values using bitwise operations, i.e.

var i = 0xFF0110, // 16711952
    rgb = {
        r: (i >> 16) & 0xFF,        // or `(i & 0xFF0000) >> 16`
        g: (i >>  8) & 0xFF,        // or `(i & 0x00FF00) >>  8`
        b:  i        & 0xFF         // or ` i & 0x0000FF       `
    }; // {r: 255, g: 1, b: 16}, same as before