Unrestricted submarine warfare

A way of saying that German submarines shoot first and ask questions later. They will sink any ship without worrying about who is on it, or which nation it belongs to.

Neutral

Meaning not taking sides in the war and not in an alliance with any other country. Germany wanted the U.S. to be more like Switzerland and less like every other country in Europe.

Alliance

One of the major causes of World War I. Nations who signed treaties of alliance were obligated to go to war for their allies, even if they thought the war was dumb and pointless, which in the case of World War I, it was. Yet, everyone got involved anyway because, alliances.

Make war together

The "war" referenced here is World War I. A war with such pointless causes that nations literally wrote each others formal letters inviting one another to join in. Like a birthday party except with millions of soldiers dying in trenches.

Lost territory in Texas, New Mexico, and Arizona

This refers to some of the land Mexico gave up to the U.S. at the end of the Mexican-American War in 1848, like, oh, most of California, Utah, Nevada, Arizona, and New Mexico. The assumption is that Mexico might like them back.

Settlement details

A concise way of telling Mexico that they can actually take as much land or whatever they want from the U.S., and Germany won't get in their way. Basically, it's up to them how much they want to make the Americans suffer.

The President

Zimmermann is referring to the President of Mexico. It would have been nice if he'd said that, but with telegrams, you're paying by the word.

Japan

Technically allied with England before this telegram, they were a militaristic state and itching to become an imperialist power. Germany took a shot at inviting them to attack the U.S.

Compelling England in a few months to make peace

When Zimmermann says, "make peace," what he really means is "surrender".

Zimmermann

The German Foreign Secretary, a position similar to the American Secretary of State, who communicates messages to ambassadors in each country often to be relayed to the heads of state. Because of this particular telegram, Zimmermann goes down in history as being not very good at his job. And the British cryptographers are remembered as being excellent at theirs.