What were relations like between Mexico and the Confederate States of America?

The answer to this is slightly complicated. From 1857, Mexico had been embroiled in its own civil war, with Conservatives (essentially landowners, the Catholic church and some indigenous communities) on one side and Liberals (bourgeoisie) on the other. The Liberals under President Benito Juarez were recognized by the United States in 1859. In December 1861, the French Second Empire under Napoleon III, together with Britain and Spain, saw their chance as the USA was distracted by their own civil war and invaded Mexico with the pretext of recovering some debts, as Juarez had suspended interest payments due to the war. Britain and Spain were satisfied by Mexican concessions and withdrew their troops but the French sought to turn Mexico into a client state and with the assistance of Mexican Conservatives they managed to occupy the capital in June 1863 and, again with the Conservatives' enthusiastic acquiescence, imposed a Habsburg prince as Emperor Maximilian I of Mexico.

The Confederacy sought recognition by and eventual alliance with the Second Mexican Empire but they were rebuffed by Maximilian, who was ironically personally quite liberal by the standards of the time and had little sympathy for the Confederates' cause. There was also little advantage to be gained by recognizing the Confederacy and a real danger of antagonizing the USA, who would remain the major regional power no matter who won the US Civil War. Juarez and the Liberals were hostile to the Confederacy throughout the period. Nevertheless, trade was conducted over the border, especially through Brownsville/Matamoros (controlled by the Empire), from where cotton and other Confederate goods were transshipped to be exchanged for arms and other essential European goods.

With the end of the US Civil War, the French withdrew their troops from Mexico in 1866 and Austro-Hungary was warned off from assisting Maximilian, who nevertheless refused to abandon the Imperial crown and held on until he was captured and executed in June 1867.