1)NO is used between two nouns indicating that the first is possessing or modifying the second.
Eg. Kore wa Yamada-san no kasa desu. This is Mr. Yamada's umbrella.
If the context is present, you can ommit the second noun.
Eg. Dare no kasa desu ka? Who's umbrella is it?
Yamada-san no desu. It's Mr. Yamada's (one).
Eg. Tsukue no ue. Top of the desk. (lit. the desk's top)
Eg. Eigo no Sensei. English teacher. (lit. The English languages's teacher)
4)As a nominaliser
Eg. Kare ga hoshii no wa, atarashii kuruma desu. What he wants is a new car. (lit. As for his wanted thing, it's a new car.)
5)at the end of sentences:
Indicates a question: (colloquial)
Eg. Nani o itte iru no? What are you saying? (You tend to hear that in anime a lot.)
6)Indicates a mild command:
Eg. Sonna koto iwanai no. Don't say things like that.
NI is use like how Sayaka said, but in addition:
1)Ni indicates destination to sth.
Eg. Ashita wa, ginko ni ikitai! I want to go to the bank tomorrow.
2)Indicates receiver of an action.
Eg. Tomodachi no tanjoubi ni wa, purezento ni ageru. I give presents to my friends on their birthday.
3)Indicates the result of change:
Eg. Jon-san wa isha ni natta. John became a doctor.
4)Indicates something in existence when used with "natte iru".
Eg. Kono tatemono no migigawa ga kyoushitsu ni natte imasu. This building's right side is a classroom.
5)Indicates a pair of things commonly mentioned together.
Eg. Romeo ni Jurietto. Romeo & Juliet. Think of the ni in this case as an ampersand &
6)In writing, ni may be used to join noins (usually 3 or more)
Eg. Soko ni, Furansu-jin ni, Igirisu-jin ni, Nihon-jin ga arimashita. There were French people, English people and Japanese people there (the 1st ni is used to indicate location and could have been followed by wa)
The simplest Japanese sentence structure is Subject-comment.
Eg. Eigo wa muzukashii. English is difficult.
However, in informal situations, someone might say "Atsui!" to mean "It's hot".
It is hot vs. Atsui.
In English there always has to be a subject (It) and a predicate (is hot.). However in Japanese the subject is inferred and the copula (to be verb) "desu/da/deshita/datta" is omitted. (in informal speech)
The point I suppose I'm trying to make is that lengthy sentences in English can be expressed with just a few words in Japanese.
There are many different sentence structures in Japanese, for example one involving adjectives, ones involving adverbs, different constructions for verbs, etc. However, the Rocket Japanese course does a good job of introducing quite a few of the frequent ones.