You might also want to be familiar with where "desu" comes from, because it takes many hidden shapes. It stems from the expression "de aru". "De" is a partical meaning "at/as", and "aru" is a verb meaning "to be/to exist". So if you wrap your brain around that, you can understand something like "to exist as... / to be in the state of..." In other words: "to be".
"Aru" is the actual verb, and the forms of "desu" reflect it. "De aru" becomes "de atte" (generally contracted to "datte") and "de atta" (contracted to "datta"). When you begin to study keigo (honorifics and levels of speech), you will learn that another common form of "desu" is "de gozaimasu" ("gozaimasu" = a powered-up form of "aru")
Used just like "desu".
友達は大切なのだ。 Tomodachi wa taisetsu na no da.
Friends are important.
-te formdeshite (rare, very formal) / datte, de
Kind of has the words "and", "because", or "therefore" built right in to it.
構わないで、好きだけ言ってくれ。 Kamawanai de, suki dake itte kure.
I don't care, [so] say what you like.
絵がすごく綺麗だって欲しくなった。E ga sugoku kirei datte hoshiku natta.
The picture was so pretty, I wanted it.
*note: Don't confuse "datte" with "da tte". While, because there are no spaces in Japanese writing, you will not necessarily see the difference, "datte" is a form of "desu", and "da tte" is sort of a verbal quotation mark.
-ta formdeshita / datta
Used for the past tense.
ひとつしかなかったから、大事だった。 Hitotsu shika nakatta kara, daiji datta.
Because I had only one, it was special.
遅くなっては失礼でした。 Osoku natte wa shitsurei deshita.
It was rude of me to be late.
Kind of has the words "like" or "sometimes" or "will" built in to it.
沙良子は、歌声が最高だったり最低だったりするよね。 Sarako wa, utagoe ga saikou dattari saitei dattari suru yo ne.
Sarako's voice will be awesome one minute, and sucky the next.
potential form:deshou / darou This form in other verbs means "let's" or "shall". It's kind of a supposing future tense, and is very common with "desu". "Deshou" > "desu" (formal), "darou" > "da" (informal).
明日は天気が晴れるでしょう。 Ashita wa, tenki ga hareru deshou.
Tomorrow the weather will [probably] be sunny.
Keita wa okoru darou. Keita will be mad [won't he?].