For the US Army, it's a bronze oak leaf. The same insignia is used by Majors in the Air Force and Marine Corps, and by those holding the rank of Lieutenant Commander in the Navy and Coast Guard while wearing certain uniforms.
The hat insignia and rank insignia contain the eagle from the great seal
The black leaf insignia is used for the Major rank in the Army. It is all for the Major rank in the Air Force and Marines.
background is scarlet and white with yellow trim and sma insignia in the middle
There is no US Army Sergeant Major badge. There is a rank insignia, but it is not an awarded badge. There are three different insignia for the rank of Sergeant Major (E-9). Sergeant Major - Three stripes on top and three stripes on bottom with one star in the middle. Command Sergeant Major - Three stripes on top and three stripes on bottom with one star and wreath and in the middle. Sergeant Major of the Army - Three stripes on top and three stripes on bottom with two stars and eagle in the middle. See related links for pictures of US Enlisted rank insignia.
Although there are different types of military badges for the US Army, there are no subdivisions for military insignia. The total number of military insignia for the US Army is 57.
Here's the list of commissioned officers for the US Army: General of the Army (5 stars), General (4 stars), Lieutenant General (3 stars), Major General (2 stars), Brigadier General (1 star), Colonel (Eagle), Lieutenant Colonel (Silver Leaf), Major (Gold Leaf), Captain (2 Silver Bars), First Lieutenant (Silver Bar), Second Lieutenant (Gold Bar).
The flag was taken from the insignia created in 1966 to distinguish the SMA rank. The flag of the Sergeant Major of the Army incorporates a silver five-pointed star surmounted by the Coat of Arms of the United States between two white five-pointed stars at the top and two red five-pointed stars at bottom. The historical rank of Sergeant Major General (which eventually became the current rank of Major General) was a 2 star position. Also - the standard insignia for a sergeant major (E-9) is three chevrons above three arcs with a (single) five pointed star between the chevrons and arcs. Sergeant Major of the Army (who is the senior representative of all enlisted soldiers) is also an E-9 but gets his own special insignia - with two stars. Originally it was just the two stars but in 1994 the insignia for Sergeant Major of the Army was changed to add the Coat of Arms of the United States between the two stars in the center of the insignia. There are thus two precedents for the use of a pair of stars for the Sergeant Major of the Army. Note that the Office of the Chief of Staff, United States Army and Office of the Vice Chief of Staff, United States Army also use star pairs on their Positional colors. Both of these have the insignia of the General Staff, a five-pointed white star, surmounted by the Coat of Arms of the United States.
A shield with an eagle in the center
There are a bunch of them. Branch of Service insignia, rank insignia and other insignia that aren't unit-specific are found in Army Regulation 670-1, which is downloadable from a few places. Distinguished Unit Insignia, which everyone calls unit crests, and patches are not in that reg; check the Institute of Heraldry website at http://www.tioh.hqda.pentagon.mil/
If the SMA is on your post, you will know it. Either the entourage surrounding him or someone within sight is going to give that away. Otherwise, the rank insignia.
Lieutenant Colonel insignia is identified by a Silver Maple Leaf. In the US Army it is a cluster of silver OAK leaves. In the British army it is a crown and a single 'pip'.
On the left lapel of the wearer.