These are basic terms, and will often crop up in forums. I find knowing the ‘naughty' terms helps for purposes of avoidance tactics (though I purposefully avoid telling some individuals ^_^).
This is a compilation of some of the anime that I have really enjoyed over the years. I have been watching anime now for over half my life. The first I ever saw was Tekkaman, followed by My Neighbor Totoro. While most of these are easily findable and hardly rare, they are great for those looking to get into the anime experience (or parents who have absolutely no idea what their kids are talking about).
10 Anime That You Absolutely Must See:
The Films of Hayao Miyazaki

These are my main arguments for the concept of anime as art. My Neighbor Totoro was one of the first anime I had ever seen. The tape kicked the bucket long since, but I still fervently follow Miyazaki. He is a truly phenomenal artist. The two that I recommend above all others are Spirited Away and Totoro. Howl's Moving Castle is significant as well, and is especially fun when you realize Christian Bale is the voice actor of the main hero. As for Totoro...well, look at him! How can you not love the giant raccoon/squirrel/panda hybrid? The Films of Makoto Shinkai

This man is a genius. He has been hailed by many as the next Miyazaki, and I fully agree with them. His first breakthrough, Voices of a Distant Star, is a sentimental, heart tugging piece. It becomes more impressive upon the discovery that he did the entire piece on his Macintosh home computer, with only himself and his fiancée as the voice actors and his friend composing the music. I especially recommend 5 Centimeters per Second. It is a down to earth, brilliant story of love, friendship, and separation. It also has the dubious honor of being one of the few anime that I have written a fan fiction for.Mobile Suit Gundam

Gundam has become almost eponymous with the concept of a large human robot with large guns and a beam sword. The original work, Mobile Suit Gundam, is a decided classic of the anime world. It has since seen several sequels and spinoffs. The first set I watched was New Mobile Report: Gundam Wing, which I still feel is one of the better of the spinoff titles. The only other spinoff that I have had much fun with was Mobile Suit Gundam SEED. I, along with many fans, site this as possibly the best spin off of the series. Gundam SEED is the other anime with a dubious honor of being good enough, yet incomplete enough that I wrote a fan fiction for it. The most recent title is Gundam 00, which, aside from sounding a little too James Bond, is one of the more favorably received, with a second season currently in the works. I recommend beginning with Mobile Suit Gundam, unless you find the animation too old fashioned, in which case I cannot recommend Gundam SEED highly enough.
The Melancholy of Haruhi Suzumiya

This series is often one of the first recommended on internet forums, and for good reason. The anime is a good example of a series that does it right. The humor is witty, the story is compelling and quirky, and the characters are all wonderfully unique. It is not serious in any real way, though it can fool you at times. This is another anime that experimented, in this case with the Light Novel, a short novel with pictures added in for effect. This is an amazing series and I agree with people who say to watch this when hunting for a new anime.Last Exile

I'm not sure how popular this is on the net, but it was one of my personal favorites. The setting is unique, something that most anime don't strive for, in a steampunk world of airships and a war that has continued for too long. The characters are interesting, the plot is compelling, and the animation and the way CG and hand drawing works together in this series is well worth the purchase.
Rurouni Kenshin and Samurai X

I class these as one set because they are all really just one long series. Trust and Betrayal are the initial story, then Rurouni Kenshin, and finally Reflections. This is the series if you want sword action. But it's more than that. It is a real look at the quality of the human condition and how far one man would go to seek redemption. It is a beautifully told story, with the action complementing the story rather than being the focus.Gankutsuou:The Count of Monte Cristo

This is a series which I picked up on a whim and fell in love with. The story of the Count is retold in a space opera setting. Before you think "what the hell," pick up at watch the first episode. The series has a few hallmarks, including a unique animating style, a decently accurate retelling of the Count of Monte Cristo, and despite the fact that you can likely see the plot twists coming, they will still throw most people for a loop.
Ghost in the Shell

This is one of those anime that takes a while to get your brain around. Quite aside from being complex, it is a real revolutionary piece. One of the classical dystopian stories of control, human advancement, and the relation of humanity and technology. Note that the movie does have spattered periodic nudity as the main heroine's cloak doesn't work through clothes... Other works by Masamune Shirow are likewise quite good. Appleseed is another one that I would recommend if this movie fails to confuse you.Fate/Stay Night

This is a wonderful anime based off of a visual novel. The story is compelling and complex, with characters that I absolutely fell in love with. It has a great blend of humor, action, romance, and drama. It is something of a cult hit (the cult being anime lovers.). Absolutely brilliant.Final Fantasy 7: Advent Children

Where do I begin? This is the sequel to the game Final Fantasy 7 for the Playstation. It is a completely original work and it took years to develop it. It has the best CG animation this side of Pixar and the characters are amazing. The fight scenes are also spectacular.