Da (だ) is a word that is used in Japanese to declare something as being. It is similar to the English words 'is, am, are', but it doesn't always translate as this. This can be used on both nouns and adjectives, but for now lets take a look at examples of it used with a noun.
Is a pen = pen da (ペンだ)
Is a student = gakusei da (学生だ)
The sentences above can change their meaning depending on the context of the situation (there is a student, this is a student, I am a student). Even thought it doesn't actually say these things it implies them. You will find a lot of times in Japanese the meaning of the sentence depends on the situation.
(I'm) a woman = onna da (女だ)
(I'm) a man = otoko da (男だ)
The negative form of da is ja nai (じゃない). This is used to show that someone or something is not.
(I'm) not Tom = tomu ja nai (トムじゃない)
The past form is datta (だった). The negative past form is ja nakatta (じゃなかった).
Was a friend = tomodachi datta (友達だった)
Wasn't a friend = tomodachi ja nakatta (友達じゃなかった)
Desu is a polite word that is often called the polite version of Da. The u on the end is silent or whispered. Both words are sometimes used in the same way, but are not always interchangeable.
Sometimes desu is used only to add politeness to the sentence. When it is used beside an adjective it makes what you are saying more polite.
Desu is not a verb but it acts like a verb and has different conjugations. The negative form of desu is the same as da (ja nai) except you add desu to the end.
Isn't a teacher = sensei ja nai desu (先生じゃないです)
The past form is deshita (でした). Negative past is nakatta desu (なかったです)
Was a book = hon deshita (本でした)
Wasn't a pen = pen ja nakatta desu (ペンじゃなかったです)
There is also another very polite negative form of desu. This form is used in very formal situations. To create this form you replace nai desu with arimasen.
Wasn't grass = kusa ja arimasen (草じゃありません)
Wasn't a TV = terebi ja arimasen deshita (テレビじゃありませんでした)
Everywhere you see the word ja (じゃ) you can also use dewa (では). Ja and Dewa mean the same thing. The difference is ja is often used in spoken Japanese, Dewa is more polite and used more in written Japanese.
Depending on what you are saying, in a lot of casual situations you can remove desu and da all together. You have to be careful on where you remove this because it can make the sentence vague or rough sounding. As you go though the lessons you will understand how to use these words more.