State of being with だ and です

Let’s start from sentences in the shape of “A is B”. Japanese has two basic "is" words. The grammar term for such words is "copula". Copulas have a dedicated term because they're special verb-like things.

The two copulas in Japanese are だ and です. です is more polite than だ. Despite both being copulas, they can't always be used in the same patterns. Funnily enough, you can even omit them entirely while still implying an “A is B” type of relationship in your sentences. Especially だ is usually omitted at the end of conversational sentences in modern Japanese.

Unlike English, the state-of-being word goes after the word that you're using to describe something, rather than before it. We will see that this applies to most modifier words in Japanese: they apply to the word on their left.

ペンだ。
It's a pen.

ネコです。
It is a cat.

The difference between だ and です is generally a matter of politeness and formality. One thing important to remember is that, while です is the polite version of だ, the two aren’t always interchangeable and have a lot of different uses. You will learn them later.

Later you will also see です get used in places where English wouldn't allow "is". This is because it can even act just like a filler word with no added meaning.

To state that something is not, we turn the だ into ではない and です into ではありません. In speech the では gets slurred into じゃ so じゃない and じゃありません are more common. We will look at what this では does in a later lesson. There exists an alternative to ではありません which is just adding です after ではない/じゃない, the meaning is the same.

人じゃない
It’s not a person

本じゃないです or 本ではありません 
It is not a book (polite)

Japanese has two basic tenses: the plain tense and the past tense. Dictionaries list verbs by their plain tense, hence it’s also called “dictionary form”. In this context, “form” and “tense” are interchangeable.

The past form of だ is だった. The past form of です is でした.

オレだった！ 
It was me! (casual)

そうでした
That was so. (polite)

The dictionary form is usually called the present tense or the non-past tense. It's important to remember that it can be used in a lot of situations. Present tense and non-past tense are reasonable names, but don't treat names like rules, sometimes names are confusing. This guide will be using these names interchangeably.

The plain form and past tense usually line up with the English present and past tenses, but they're not used in all the same places. Just remember that English and Japanese often use tenses in very different ways.

In Japanese, ordinary statements about the future use the plain form most of the time. This is where the name "non-past" comes from.

Sometimes the past tense can also be used for things that are already completed like "has become", even if they're hypothetical and haven't happened yet. You might also hear it called “completed tense” or “completed form”.