Their names should help you:

• android:gravity sets the gravity of the contents (i.e. its subviews) of the View it's used on.
• android:layout_gravity sets the gravity of the View or Layout relative to its parent.

And an example is here.

Inside - Outside

• gravity arranges the content inside the view.
• layout_gravity arranges the view's position outside of itself.

Sometimes it helps to see a picture, too. The green and blue are TextViews and the other two background colors are LinearLayouts.

Notes

• The layout_gravity does not work for views in a RelativeLayout. Use it for views in a LinearLayout or FrameLayout. See my supplemental answer for more details.
• The view's width (or height) has to be greater than its content. Otherwise gravity won't have any effect. Thus, wrap_content and gravity are meaningless together.
• The view's width (or height) has to be less than the parent. Otherwise layout_gravity won't have any effect. Thus, match_parent and layout_gravity are meaningless together.
• The layout_gravity=center looks the same as layout_gravity=center_horizontal here because they are in a vertical linear layout. You can't center vertically in this case, so layout_gravity=center only centers horizontally.
• This answer only dealt with setting gravity and layout_gravity on the views within a layout. To see what happens when you set the gravity of the of the parent layout itself, check out the supplemental answer that I referred to above. (Summary: gravity doesn't work well on a RelativeLayout but can be useful with a LinearLayout.)

So remember, layout_gravity arranges a view in its layout. Gravity arranges the content inside the view.

xml

Here is the xml for the above image for your reference:

<?xml version="1.0" encoding="utf-8"?><LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"    android:layout_width="match_parent"    android:layout_height="match_parent"    android:orientation="vertical" >    <LinearLayout        android:layout_width="match_parent"        android:layout_height="0dp"        android:layout_weight="1"        android:background="#e3e2ad"        android:orientation="vertical" >        <TextView            android:layout_width="wrap_content"            android:layout_height="wrap_content"            android:layout_gravity="center_horizontal"            android:textSize="24sp"            android:text="gravity=" />        <TextView            android:layout_width="200dp"            android:layout_height="40dp"            android:background="#bcf5b1"            android:gravity="left"            android:text="left" />        <TextView            android:layout_width="200dp"            android:layout_height="40dp"            android:background="#aacaff"            android:gravity="center_horizontal"            android:text="center_horizontal" />        <TextView            android:layout_width="200dp"            android:layout_height="40dp"            android:background="#bcf5b1"            android:gravity="right"            android:text="right" />        <TextView            android:layout_width="200dp"            android:layout_height="40dp"            android:background="#aacaff"            android:gravity="center"            android:text="center" />    </LinearLayout>    <LinearLayout        android:layout_width="match_parent"        android:layout_height="0dp"        android:layout_weight="1"        android:background="#d6c6cd"        android:orientation="vertical" >        <TextView            android:layout_width="wrap_content"            android:layout_height="wrap_content"            android:layout_gravity="center_horizontal"            android:textSize="24sp"            android:text="layout_gravity=" />        <TextView            android:layout_width="200dp"            android:layout_height="40dp"            android:layout_gravity="left"            android:background="#bcf5b1"            android:text="left" />        <TextView            android:layout_width="200dp"            android:layout_height="40dp"            android:layout_gravity="center_horizontal"            android:background="#aacaff"            android:text="center_horizontal" />        <TextView            android:layout_width="200dp"            android:layout_height="40dp"            android:layout_gravity="right"            android:background="#bcf5b1"            android:text="right" />        <TextView            android:layout_width="200dp"            android:layout_height="40dp"            android:layout_gravity="center"            android:background="#aacaff"            android:text="center" />    </LinearLayout></LinearLayout>

Related

• Difference between a View's Padding and Margin
• Match_parent vs wrap_content
• How to set both gravity and layout gravity of a LinearLayout programatically

The difference

android:layout_gravity is the Outside gravity of the View. Specifies the direction in which the View should touch its parent's border.

android:gravity is the Inside gravity of that View. Specifies in which direction its contents should align.

HTML/CSS Equivalents

(if you are coming from a web development background)

Android                 | CSS————————————————————————+————————————android:layout_gravity  | floatandroid:gravity         | text-align

Easy trick to help you remember

Take layout-gravity as "Lay-outside-gravity".