State Arms of Mexico

The text of the Law on the Arms and the Flag of Mexico is available in Spanish, with this illustration of the arms. The law provides that:
"El Escudo Nacional esta constituido por un águila mexicana, con el perfil izquierdo expuesto, la parte superior de las alas en un nivel mas alto que el penacho y ligeramente desplegadas en actitud de combate; con el plumaje de sustentación hacia abajo tocando la cola y las plumas de esta en abanico natural. Posada su garra izquierda sobre un nopal florecido que nace en una pena que emerge de un lago, sujeta con la derecha y con el pico, en actitud de devorar, a una serpiente curvada, de modo que armonice con el conjunto. Varias pencas del nopal se ramifican a los lados. Dos ramas, una de encino al frente del águila y otra de laurel al lado opuesto, forman entre ambas un semicírculo inferior que se unen por medio de un listón dividido en tres franjas que, cuando se representa el Escudo Nacional en colores naturales, corresponden a los de la Bandera Nacional."

Unlike the Maya, the Mexica (as they called themselves, the word Aztec were invented much later to denote the real Mexica, as opposed to those who happen to live within the modern country of Mexico) did not have a complete system of writing. What they did have was a system of glyphs. There were name glyphs, and there were glyphs for many actions and things. There was e.g. one accepted way of drawing a temple. A burning temple meant defeat. The name glyph of Tlatelolco next to a burning temple would mean "Tlatelolco is conquered", or "the defeat of Tlatalolco" and similar meanings.

The name glyph of Tenochtitlan was a cactus issuant from a stone. (Tenochtitlan was the capital of the Mexica. Mexico City includes the sites of the ancient cities of Tenochtitlan, Tlatelolco, and Texcoco). The arms of Mexico is based on the Tenochtitlan name glyph, though it doesn't look quite the same. The "stone" glyph, which looks rather like the central part of a heraldic thunderbolt, is surmounted by an eagle holding in one claw a serpent.

The glyph refers to the legend that the site of the city was chosen when and eagle landed on a cactus. Tenochtitlan means something like cactus fruit place.

The glyph appears in pre-Columbian documents, and also becomes common in the 18th century. It was adopted as National arms from the very beginning, on April 9, 1823.

Arms of the City of Mexico

On July 4, 1523, Carlos V of Spain granted arms to the City of Mexico. It can be blazoned as follows: Azure a castle between three bridges or, two from each flank and one from the base, converging toward the castle but not touching it; on each the two bridges from the flank a lion rampant of the last supporting the castle; on a bordure or twelve leaves of the nopal cactus vert. The Spanish text of the royal Cedula follows:
"un escudo azul con un castillo dorado en medio y tres puentes de piedra de cantería en que van a dar a dicho castillo, los dos sin llegar a él, y a cada uno de los dichos duos puentes, que han de estar a los lados, un león levantado, que asga con las uñas del dicho castillo [...] de manera que tenfan las patas en el puente y los brazos en el castillo en señal de victoria que en ella vieron los cristianos [...] y por orla diez hojas de tuna verdes con sus abrojos que nacen en la dicha provincia en campo dorado a un escudo tal como este."

Over the course of time these arms were modified: an eagle was added on top of the castle, the lions disappeared, the eagle began to hold a serpent, the cactus leaves migrated from the bordure to the castle. In other words, the coat of arms transmogrified into the pre-Columbian glyph.

Arms of the Empire of Mexico

See Casa Imperial, a site devoted to the Mexican monarchy.

The (2nd) Empire of Mexico lived only briefly: a "Regencia del Imperio" was proclaimed in 1863 with the arrival of French troops, and Maximilian of Austria accepted the Imperial crown of Mexico in 1864. The empire ended in 1867, with the execution of Maximilian. Its arms were quite similar to the preceding and following regimes:

Azure, from a sea argent a rock issuant proper thereon a cactus (Nopal) proper, on the cactus an eagle or holding in its beak and claws a serpent vert. The oval shield is rimmed with a bordure or with laurel branches, it is surrounded by the collar of the order of St.Maria of Guadalupe (until 1865) or the order of the Mexican Eagle (after 1865), the scepter and sword are placed per saltire behind the shield. The shield is crowned with an imperial crown of eagles (similar to the French imperial crown: the eagles' wings curve back to form the arches of the crown and are all joined at the top to support a fleur-de-lys). The supporters are two Austrian griffons (per fess sable and or), and the motto is "Equidad en la justicia".

The Regencia adopted similar arms (the crown above the shield is a traditional one with strawberry leaves and an orb, the eagle is crowned) on November 20, 1863. Maximilian adopted his arms by a decree of July 18, 1864 whose text does not describe the arms (a depiction was attached). Another decree of November 10, 1865 describes the arms as follows:
"El escudo de armas del Imperio es de forma oval y campo azul; lleva en el centro el águila de Anáhuac de purfil pasante sostenida por un nopal, soportado por una roca inundada de agua, y desguarando la serpiente; la bordura es de oro, congada de los ramos de encina y laurel, timbrado con la corona imperial; por soportes tiene los dos grifos de las armas de Nuestros mayores, mitad, la parte superior negra y la inferior de oro; y por en sotuer el cetro y la espada: está rodeada del collar de la Orden del Aguila Mexicana, y por divisa "Equidad en la Justicia"."

Reference: Siebmacher. Manuel Carrera Stampa, El Escudo Nacional, Mexico, 1960.