Leaflet.draw retrieve layer type on draw:edited event

You could use instanceof (docs here).

map.on('draw:edited', function (e) {
    var layers = e.layers;
    layers.eachLayer(function (layer) {
        if (layer instanceof L.Marker){
            //Do marker specific actions here
        }
    });
});

Be very careful when using instanceof. Leaflet.draw plugin uses standard Leaflet's L.Rectangle. Leaflet's rectangle extends Polygon. Polygon extends Polyline. Therefore, some shapes might give you unexpected results using this method.

var drawnItems = new L.FeatureGroup();
map.addLayer(drawnItems);

... add layers to drawnItems ...

// Iterate through the layers    
drawnItems.eachLayer(function(layer) {

    if (layer instanceof L.Rectangle) {
        console.log('im an instance of L rectangle');
    }

    if (layer instanceof L.Polygon) {
        console.log('im an instance of L polygon');
    }

    if (layer instanceof L.Polyline) {
        console.log('im an instance of L polyline');
    }

});

How do i find out the shape type?

var getShapeType = function(layer) {

    if (layer instanceof L.Circle) {
        return 'circle';
    }

    if (layer instanceof L.Marker) {
        return 'marker';
    }

    if ((layer instanceof L.Polyline) && ! (layer instanceof L.Polygon)) {
        return 'polyline';
    }

    if ((layer instanceof L.Polygon) && ! (layer instanceof L.Rectangle)) {
        return 'polygon';
    }

    if (layer instanceof L.Rectangle) {
        return 'rectangle';
    }

};