だ and です are the same verb. Both mean "to be" in English, but だ is informal while です is formal (see Verb Conjugation for more on Japanese levels of politeness). だ・です is a copula, or linking verb and cannot be used by itself. It is primarily used to state, affirm, or assert (something) about the subject of a sentence.

The Conjugation

Like the English verb "to be", conjugation of だ・です is irregular when compared to other verbs:

普通語 Informal
is/am/are[Present Affirmative]
ではない* is/am/are not[Present Negative]
だった was/were[Past Affirmative]
ではなかった* was/were not[Past Negative]
だろう is/am/are probably[Tentative]
丁寧語 Polite
です is/am/are[Present Affirmative]
ではありません* is/am/are not[Present Negative]
でした was/were[Past Affirmative]
ではありませんでした* was/were not[Past Negative]
でしょう is/am/are probably[Tentative]
Conditional and Conjunctive
だったら・であれば if ... is/am/are[Conditional]
ではなかったら・ではなければ* if ... is/am/are not[Negative Conditional]
is/am/are ... and/so[Conjunctive]
ではなくて* is/am/are not ... and/so[Negative Conjunctive]
* a more colloquial form of では is じゃ so: じゃない = ではない

Note that the English equivalents for both the informal and polite tables above are identical. Which to use depends entirely on the level of politeness you wish to express, based on your relation with the person you are speaking to.

Some Example Sentences

Here are some example sentences using the above conjugations of だ/です. Note that there is no conjugation for future tense.  In Japanese the present tense is used for future tense and, when it is not already known by the listener, the future date or time is referenced by the speaker.  Often the tentative form is used when referring to the future.**

Informal Formal
彼は泳者だ。
[kare wa eisha da.]
He is a swimmer.
彼は泳者です。
[kare wa eisha desu.]
He is a swimmer.
彼は泳者ではない。
[kare wa eisha dewa nai.]
He is not a swimmer.
彼は泳者ではありません。
[kare wa eisha dewa arimasen.]
He is not a swimmer.
彼は泳者だった。
[kare wa eisha datta.]
He was a swimmer.
彼は泳者でした。
[kare wa eisha deshita.]
He was a swimmer.
彼は泳者ではなかった。
[kare wa eisha dewa nakatta.]
He was not a swimmer.
彼は泳者ではありませんでした。
[kare wa eisha dewa arimasen deshita.]
He was not a swimmer.
彼は泳者だろう。
[kare wa eisha darou.]
He is probably a swimmer.
彼は泳者でしょう。
[kare wa eisha deshou.]
He is probably a swimmer.
彼は泳者だったら元気だろう。
[kare wa eisha dattara genki darou.]
If he is a swimmer, he is probably healthy.
彼は泳者だったら元気でしょう。
[kare wa eisha dattara genki deshou.]
If he is a swimmer, he is probably healthy.
彼は泳者で上手に泳げる。
[kare wa eisha de jouzu ni oyogeru.]
He is a swimmer, so he swims well.
彼は泳者で上手に泳げます。
[kare wa eisha de jouzu ni oyogemasu.]
He is a swimmer, so he swims well.
彼は明日もまだ泳者だろう。**
[kare wa ashita mo mada eisha darou.]
He'll probably still be a swimmer tomorrow.
彼は明日もまだ泳者でしょう。**
[kare wa ashita mo mada eisha deshou.]
He'll probably still be a swimmer tomorrow.

In addition to serving as a predicate, だ・です is used to determine the tense of -な adjectives, and the level of politeness for both -な and -い adjectives. It can also be used on occasion to replace certain predicate verbs as in the example: 僕は寿司だ。(I'll have sushi.) in response to the question: 何にしますか? (What will you have?).