The zero copula: no verb in the present tense
In the present tense Russian leaves out 'am', 'is' and 'are' altogether; the verb быть appears only in the past and the future.
English cannot build a sentence without a verb, so 'I am a student' needs its 'am'. Russian builds the same sentence with no verb at all: Я студе́нт. This is not casual speech or an abbreviation. It is the standard, neutral present-tense form, used in writing as well as in conversation. The verb 'to be' — быть — exists, but in the present tense it stays invisible. This single fact is worth more in your first hour than any conjugation table: before you have learnt one verb ending you can already produce hundreds of true sentences.
Leave the present tense and быть comes straight back. The past has four forms, and they agree with the subject in gender and number, never in person: был (masculine), была́ (feminine), бы́ло (neuter), бы́ли (plural). The future is conjugated like an ordinary verb: бу́ду, бу́дешь, бу́дет, бу́дем, бу́дете, бу́дут. So one idea runs Он студе́нт → Он был студе́нтом → Он бу́дет студе́нтом, and only the first of those three needs no verb. Notice the change in the last two: once быть is present, the noun that follows it normally shifts into the Instrumental case. You meet that case in Lesson 10; until then, stay in the present or use an adjective.
When both halves of the sentence are nouns in the nominative, Russian usually writes a dash where the missing verb would be: Москва́ — го́род. Мой брат — врач. Кни́га — э́то пода́рок. The dash is punctuation, not a word, and in speech it corresponds to a short pause. It disappears in three situations you will meet constantly: when the subject is a personal pronoun (Я студе́нт, Она́ врач), when the predicate is negated (Он не врач), and in a question (Ты студе́нт?). It also disappears when the predicate is an adjective rather than a noun (Пого́да хоро́шая).
There is a word есть, which is historically the present tense of быть, and it survives — but with a different job. Meaning 'there is / there are' it is completely ordinary: У нас есть вре́мя, В го́роде есть парк. Used as a plain copula (Я есть студе́нт) it is emphatic, the way English stresses 'I AM a student'. Its negative is not *не есть but нет plus the Genitive: У нас нет вре́мени.
Examples
- Я студе́нт. I am a student. — no verb at all Listen in Russian: Slow in Russian: Я студе́нт.
- Москва́ — столи́ца Росси́и. Moscow is the capital of Russia. — a dash stands where 'is' would be Listen in Russian: Slow in Russian: Москва́ — столи́ца Росси́и.
- Он не врач, он инжене́р. He is not a doctor, he is an engineer. — no dash after не Listen in Russian: Slow in Russian: Он не врач, он инжене́р.
- Вчера́ она́ была́ до́ма. Yesterday she was at home. — быть returns in the past Listen in Russian: Slow in Russian: Вчера́ она́ была́ до́ма.
- За́втра мы бу́дем в Лондо́не. Tomorrow we shall be in London. Listen in Russian: Slow in Russian: За́втра мы бу́дем в Лондо́не.
- У нас есть вре́мя. We have time. (literally: at us there is time) Listen in Russian: Slow in Russian: У нас есть вре́мя.