Learn Russian: Наш класс большой.

Possessive pronouns like "our" must match the gender of the noun they describe. The word класс ends in a consonant, which makes it a masculine noun. Therefore, we use the masculine form наш. If we were talking about a feminine noun, we would use наша.

Where is the word for "is" in this sentence?

In Russian, the present tense of the verb "to be" (am, is, are) is almost always skipped. You just say the equivalent of "Our class big" and the "is" is automatically understood!

Why does большой end in -ой instead of the usual masculine adjective ending -ый?

Most masculine adjectives end in -ый or -ий. However, when the spoken stress (the emphasis) falls on the very last syllable of the adjective, the spelling rule changes the ending to -ой. In большой, the emphasis is heavily on that final sound, so it takes the -ой ending.

Does класс mean a group of students, or a physical classroom?

It can actually mean both, just like in English! Класс can refer to the physical room, the group of students you study with, or even the grade level. You have to rely on the context to know which one is meant.