Mexico is a federal state, and its capital is located in a special federal district.
They are somewhat short on names: The country, the capital, and one of the states all share the same name: Mexico.
The Mexicans call their capital "Ciudad de Mexico" which translates to English as "Mexico City".
Many companies here are called something-mex: The phone company Telmex, the bank Banamex and the gas station Pemex.
They use the $ sign to designate Pesos (worth about 10 cents) and sometimes the prices still look reasonable even in US dollars.
They use the letter E to designate Parking.
Mexico's major international airport in Mexico City is inside the city limits.
And finally, when they say San Francisco here, they refer to the saint, not the city, unless you're flying there, like I am.