How to create a floating action button (FAB) in android, using AppCompat v21?

There is no longer a need for creating your own FAB nor using a third party library, it was included in AppCompat 22.

https://developer.android.com/reference/android/support/design/widget/FloatingActionButton.html

Just add the new support library called design in in your gradle-file:

compile 'com.android.support:design:22.2.0'

...and you are good to go:

<android.support.design.widget.FloatingActionButton
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:layout_gravity="bottom"
        android:layout_margin="16dp"
        android:clickable="true"
        android:src="@drawable/ic_happy_image" />

I've generally used xml drawables to create shadow/elevation on a pre-lollipop widget. Here, for example, is an xml drawable that can be used on pre-lollipop devices to simulate the floating action button's elevation.

<?xml version="1.0" encoding="utf-8"?>
<layer-list xmlns:android="http://schemas.android.com/apk/res/android">
<item android:top="8px">
    <layer-list>
        <item>
            <shape android:shape="oval">
                <solid android:color="#08000000"/>
                <padding
                    android:bottom="3px"
                    android:left="3px"
                    android:right="3px"
                    android:top="3px"
                    />
            </shape>
        </item>
        <item>
            <shape android:shape="oval">
                <solid android:color="#09000000"/>
                <padding
                    android:bottom="2px"
                    android:left="2px"
                    android:right="2px"
                    android:top="2px"
                    />
            </shape>
        </item>
        <item>
            <shape android:shape="oval">
                <solid android:color="#10000000"/>
                <padding
                    android:bottom="2px"
                    android:left="2px"
                    android:right="2px"
                    android:top="2px"
                    />
            </shape>
        </item>
        <item>
            <shape android:shape="oval">
                <solid android:color="#11000000"/>
                <padding
                    android:bottom="1px"
                    android:left="1px"
                    android:right="1px"
                    android:top="1px"
                    />
            </shape>
        </item>
        <item>
            <shape android:shape="oval">
                <solid android:color="#12000000"/>
                <padding
                    android:bottom="1px"
                    android:left="1px"
                    android:right="1px"
                    android:top="1px"
                    />
            </shape>
        </item>
        <item>
            <shape android:shape="oval">
                <solid android:color="#13000000"/>
                <padding
                    android:bottom="1px"
                    android:left="1px"
                    android:right="1px"
                    android:top="1px"
                    />
            </shape>
        </item>
        <item>
            <shape android:shape="oval">
                <solid android:color="#14000000"/>
                <padding
                    android:bottom="1px"
                    android:left="1px"
                    android:right="1px"
                    android:top="1px"
                    />
            </shape>
        </item>
        <item>
            <shape android:shape="oval">
                <solid android:color="#15000000"/>
                <padding
                    android:bottom="1px"
                    android:left="1px"
                    android:right="1px"
                    android:top="1px"
                    />
            </shape>
        </item>
        <item>
            <shape android:shape="oval">
                <solid android:color="#16000000"/>
                <padding
                    android:bottom="1px"
                    android:left="1px"
                    android:right="1px"
                    android:top="1px"
                    />
            </shape>
        </item>
        <item>
            <shape android:shape="oval">
                <solid android:color="#17000000"/>
                <padding
                    android:bottom="1px"
                    android:left="1px"
                    android:right="1px"
                    android:top="1px"
                    />
            </shape>
        </item>
    </layer-list>
</item>
<item>
    <shape android:shape="oval">
        <solid android:color="?attr/colorPrimary"/>
    </shape>
</item>
</layer-list>

In place of ?attr/colorPrimary you can choose any color. Here's a screenshot of the result: