Make anchor link go some pixels above where it's linked to

window.addEventListener("hashchange", function () {
    window.scrollTo(window.scrollX, window.scrollY - 100);
});

This will allow the browser to do the work of jumping to the anchor for us and then we will use that position to offset from.

EDIT 1:

As was pointed out by @erb, this only works if you are on the page while the hash is changed. Entering the page with a #something already in the URL does not work with the above code. Here is another version to handle that:

// The function actually applying the offset
function offsetAnchor() {
    if(location.hash.length !== 0) {
        window.scrollTo(window.scrollX, window.scrollY - 100);
    }
}

// This will capture hash changes while on the page
window.addEventListener("hashchange", offsetAnchor);

// This is here so that when you enter the page with a hash,
// it can provide the offset in that case too. Having a timeout
// seems necessary to allow the browser to jump to the anchor first.
window.setTimeout(offsetAnchor, 1); // The delay of 1 is arbitrary and may not always work right (although it did in my testing).

NOTE: To use jQuery, you could just replace window.addEventListener with $(window).on in the examples. Thanks @Neon.

EDIT 2:

As pointed out by a few, the above will fail if you click on the same anchor link two or more times in a row because there is no hashchange event to force the offset.

This solution is very slightly modified version of the suggestion from @Mave and uses jQuery selectors for simplicity

// The function actually applying the offset
function offsetAnchor() {
  if (location.hash.length !== 0) {
    window.scrollTo(window.scrollX, window.scrollY - 100);
  }
}

// Captures click events of all <a> elements with href starting with #
$(document).on('click', 'a[href^="#"]', function(event) {
  // Click events are captured before hashchanges. Timeout
  // causes offsetAnchor to be called after the page jump.
  window.setTimeout(function() {
    offsetAnchor();
  }, 0);
});

// Set the offset when entering page with hash present in the url
window.setTimeout(offsetAnchor, 0);

JSFiddle for this example is here

Working only with css you can add a padding to the anchored element (as in a solution above) To avoid unnecessary whitespace you can add a negative margin of the same height:

#anchor {
    padding-top: 50px;
    margin-top: -50px;
}

I am not sure if this is the best solution in any case, but it works fine for me.

Even better solution:

<p style="position:relative;">
    <a name="anchor" style="position:absolute; top:-100px;"></a>
    I should be 100px below where I currently am!
</p>

Just position the <a> tag with absolute positioning inside of a relatively positioned object.

Works when entering the page or through a hash change within page.