Appellation

A rose by any other appellation would smell as sweet. That's because appellation means "name" or "designation."

Apprise

Basically, this just means "tell," or if you're feeling fancy, "inform." For instance, "And I'm apprising you, I'm not going/You're the best man I'll ever know…" Okay, admittedly that doesn't have quite the same ring to it.

Beneficence

This is the noun version of "beneficent," which is very similar to "benevolent." It means "charity," "generosity," "kindness," etc. In other words, it's about bestowing benefits on someone else in some way. It also kind of rhymes with "Maleficent" (but that doesn't really help because she's evil and not really kind, but we mention it just for pronunciation help).

Bloody contests

Washington is reminding his readers, who probably didn't really need a reminder, that there's been a lot of conflict between other countries that the United States seems to be getting pulled into. He doesn't give a lot of detail because it was still happening at the time, and people would have gotten the reference.

Caprice

There are a few ways to use this word, but basically it means "unpredictable change." Washington uses it in reference to Europe and why the United States shouldn't get too close to any European countries—because their tendencies to change their minds and policies on a whim aren't great to deal with.

Diffidence

This kind of sounds like "difference," but really it means something more like "insecurity." So, when Washington talks about his "diffidence to myself," he's reiterating that he thinks he's slightly incompetent. You're too harsh on yourself, George.

Enfeeble

Well, "feeble" means "weak," so to enfeeble means "to weaken." (See what we did there?)

Exigencies

"Exigencies" are needs that require some kind of action. The "public exigencies" (30.4) that Washington mentions, then, refer to any general situations that people might encounter that would need government action. He's trying to convince his audience that taxes (i.e., revenue) are necessary because stuff happens that needs money to fix.

Faction

Washington uses the term "faction" to describe the divisions that had developed in the country, mostly in politics. He uses the noun version of the word, which just means a "section or group of people," or a "clique." Generally, you use the word when you're talking about rival groups that don't exactly get along. The political divisions in the United States were often along regional lines, so it also kind of refers to different parts of the country as well.

Jay Treaty

Washington actually mentions "the formation of two treaties, that with Great Britain, and that with Spain" (16.4). The one with Spain is the Treaty of San Lorenzo (see below), but the one with Great Britain was known as the Jay Treaty. It wasn't quite as successful as the San Lorenzo one. Britain had been using their naval power to restrict American trade and kidnap American sailors, and the war with France just made things worse. John Jay tried to negotiate a treaty, but he didn't have a lot of leverage, so the resulting treaty didn't really accomplish anything and was pretty unpopular (source).

Ligament

Literally, a ligament is a fibrous band of tissue that connects bones to each other or to muscles, especially at the joints. More poetically, it's just a general bond or something that ties people (or things, concepts, etc.) to each other. Guess which way Washington uses it in his farewell address.

The North/the South/the East/the West

Washington talks a bit about regional differences, and he specifically mentions the big four. But remember, in 1796, the country was a lot smaller, and those regions were defined very differently than they would be now. Take a look at a map from the time period. At the time, the "West" was the territory that would become what we now think of as part of the Midwest: Ohio, Illinois, Michigan, etc. The "South" ends east of Louisiana, which the United States got in 1803.

Odium

This is the noun version of "odious," so essentially it means "serious hatred." Washington talks about how people who pick one foreign nation over another do it "without odium" (33.4), which means no one gets upset with them for picking sides.

Solicitude

This is anxiety, concern—basically what your mom feels when you don't call her back right away.

Suffrages

"Suffrage" is the right to vote. Washington uses the plural in a kind of unusual way (at least to our ears), but really he just means "votes," or "elections."

Treaty of San Lorenzo/Pinckney's Treaty

Washington references "the treaty with Spain" (16.4). He's referencing the Treaty of San Lorenzo, a.k.a. Pinckney's Treaty, which allowed the United States to trade freely on the Mississippi River. The lower part of the river had been held by the Spanish forever, and they'd kept tight control to prevent the United States from getting too far west. With the French Revolution and the European wars that followed, though, Spain had to focus on other things and make nice with the United States, so they agreed to loosen their grip on the river route (source).

Weal

This isn't one you hear/read much nowadays. It means "prosperity" or "well-being." If you ever need a homonym for "wheel," you're got it now.