While many words have entered English directly from German, many other words simply did not. English and German are in the same Germanic language family, so the two languages share many words. In fact, oftentimes two words will be derived from the same root but mean completely different things in the respective languages. This is known as a “false cognate." A great example of this is “pickle," which is a tasty thing to garnish a burger with in English but is a pimple in German. Another is the word “gift": something you give to another in friendship in English, but poison in German!
If there are “false" cognates, then there must also be “true" cognates. These words share the same meaning and (mostly) the same spelling between two languages, here German and English. These can be derived from a variety of sources: from the explosion of Denglisch and the Anglicization of German, or the reverse, the integration of German into English. The latter category of cognates is known as “loan words," from German to English, and they will be the subject of the remainder of this article.
Admittedly, there is usually some small difference in connotation between the German version of the word and the English. It is a sliding scale between a “true" cognate and “false" one, with shades of meaning in between. The choice of words here reflects the former category more, and any differences that exist are noted.
The cause? Often, it is simply because no English word exists with exactly the same meanings and connotations that a writer or translator wishes to express, so they simply leave the word be. Add in popular and/or fashionable use of the word, and it enters everyday use in the English language.
Angst: Fear. This has developed very powerful connotations from its use in Germanic philosophy to describe a soul-deep sense of fear about the self. However, it is also used somewhat satirically to describe the often-ridiculous emotional turmoil within teenagers, “teenage angst."
Ansatz: More commonly used in academic English than in colloquial, this word has been borrowed to describe the beginning or onset of an event or idea.
Blitzkrieg: History nuts will know this one well: literally “lightning war," in particular with reference to the rapid-fire strategy of the Germans during WWII as they marched steadily across Western Europe. In English, it is also used generally to refer to any fast strategy, either in business or in war.
Bratwurst: A staple of every “American" barbecue.
Delikatessen: Often slightly Anglicized to “Delicatessen," this refers to a shop that sells delicious, tasty things. Spy the “essen" in there, which means “to eat"?
Doppelgänger: Used to describe a look-alike of somebody, often in a sinister sense.
Ersatz: This adjective is used to refer to a mediocre imitation or substitution of the “real thing," whatever that may be.
Fest: Like a party, only a festival!
Firn: A mountaineering term to refer to snow from previous seasons, often exposed in the fierce heat of summer as last winter's snow melts away.
Frankfurter: An item of delicious food, which, like the bratwurst, is a staple of “American" barbecues.
Gestalt: This term has been borrowed particularly in the fields of psychology and philosophy to refer to a single whole concept created by a collection of individual concepts. It's a bit tricky to translate this into English; hence, its direct borrowing from German.
Glockenspiel: A musical instrument where metal bars are struck with a mallet to produce sound.
Hamburger: Another food popularly considered “American," even if does take its name directly from a German city.
Hinterland: A favorite word of journalists and poets to describe the backwoods and interior of a given country, usually with reference to its lack of development.
Kaputt: Broken! Usually spelled with a single t in English, “kaput."
Kindergarten: Literally “children garden," a place for children to start in on their education while still playing around.
Meister: A master of some skill. Often used comically in English.
Poltergeist: Any kind of noisy, disruptive ghost. An excellent pop culture example in English would be Peeves the Poltergeist from the Harry Potter books.
Pretzel: Think salty, wholesome and delicious.
Rucksack: Ever wondered what the “ruck" had to do with the “sack"? Turns out, it comes from the German word for back—so it is literally just a backpack. “Rucksack" in English is often used to describe a broader category of heavy-duty bags that do not go on the back.
Sauerkraut: Tasty or not? Regardless of personal feelings on the subject, “sauerkraut" has directly entered the English language to refer to yet another boiled cabbage dish.
Über-: This prefix and occasional preposition has a certain amount of gratuitous use online and in print to refer to an excessive amount of something, the prefix literally translating as “over." Über-Übering, anyone?
Übermensch: literally “superhuman" or sometimes translated “overman." The German philosopher Nietzsche developed this concept with some political results. In pop culture, it rarely has all that much to do with his actual thoughts.
Wanderlust: “Lust" in German has somewhat different connotations than in English: a simple pleasure in something, not necessarily tinged with anything sexual. Add this to “Wander," and the meaning becomes a desire to walk and travel and sometimes obsession.
Zeitgeist: Another ghost word, literally translating as “time ghost." This is used to describe the current cultural and political trends of any day and age.
More Loan Words?
Obviously, many more words have been borrowed from German in the English language. Feel free to add some more in the comments!
Otherwise, my personal recommendation for finding more would be simply to keep your ears and eyes wide open while learning the language for things you've already seen. Loanwords are a great exercise in how culture plays into the development of a language.
There are many lists of loanwords available. Wikipedia is my personal favorite, as it has detailed articles on the exact connotations and development of each word.