Entering address details into Google Maps geocoder
The main change I would make is to reduce the responsibility of codeAddress() to do just the geocoding.  If it successfully looks up the address, then it can pass the location to a callback.

I don't have enough context to tell whether geocoder, map, and marker are global variables, or whether all of this code lies inside some other scope.  Out of habit, I've converted them into local variables.  You decide whether that's right for you.  The point variables in your drag handlers should definitely be made local, though!

function initialize() {
    var geocoder = new google.maps.Geocoder();
    var map = new google.maps.Map(document.getElementById('map-canvas'));

    var marker = new google.maps.Marker({
        map: map,
        position: map.getCenter(),
        draggable: true
    });

    google.maps.event.addListener(marker, 'dragend', function() {
        var point = marker.getPosition();
        map.panTo(point);
        showCoords(point);
    });

    google.maps.event.addListener(map, 'dragend', function() {
        var point = marker.getPosition();
        marker.setPosition(map.getCenter());
        showCoords(point);
    });

    var address = document.getElementById('address').value;

    codeAddress(geocoder, address, function(location) {
        map.setCenter(results[0].geometry.location);
        map.setZoom(15);
        marker.setPosition(map.getCenter());
        showCoords();
    });
}

function codeAddress(geocoder, address, callback) {
    geocoder.geocode({'address': address}, function(results, status) {
        if (status == google.maps.GeocoderStatus.OK) {
            callback(results[0].geometry.location);
        } else {
            alert('Geocode was not successful for the following reason: ' + status)
        }
    });
}