Learn Hardcore Japanese: asita siken ga arimasu.

In Japanese, time words like 明日 often appear at the start of a sentence without any particle. They function adverbially to set the time frame. You could add に (明日に試験があります), but it’s perfectly natural—and more common—to drop it when a time word comes first.

Why is 試験 marked with が instead of を?

The verb あります means “to exist” (or “to have,” in a sense). That makes 試験 the grammatical subject of the verb, not a direct object. Subjects in Japanese are marked by が, so you say 試験がある, not 試験をある.

What exactly does あります mean? Why not use です?

あります is the polite form of the intransitive verb ある, meaning “to exist” (for inanimate things or events).

です is the copula “to be,” used to equate or describe (e.g., 学生です “(I) am a student”).
To express that something exists (like an exam), you need ある/あります, not です.

Why can’t we say います instead of あります?

Japanese distinguishes between animate and inanimate existence:

いる/います for people and animals (animate)
ある/あります for everything else (inanimate, events, abstract things)
An exam is not animate, so you must use あります.

How do we know this sentence refers to the future?

Japanese doesn’t have a separate future tense. The same verb form can be present or future. We understand 明日 (“tomorrow”) to mean it’s an upcoming event. Without 明日, 試験があります could mean “there is an exam” (now or in general) or “there will be an exam.”

Why is が used instead of the topic marker は?

が often introduces new or important information—“there exists an exam.”
If you use は (明日試験はあります), it would sound like you’re contrasting or emphasizing “as for the exam tomorrow…” which is less natural unless you’re comparing it to something else.

Where is the “we” or “I” in “we have an exam tomorrow”? Who does the exam belong to?

In 試験があります, 試験 is the subject, and the existence verb ある doesn’t need an owner. Contextually it often implies “we have an exam.” If you want to specify the person who has it, you can add an indirect object with に:
明日私たちに試験があります (“There is an exam for us tomorrow”).

Is あります polite or plain?

Because it ends in ます, あります is the polite form. The plain (dictionary) form is ある:
Plain: 明日試験がある。
Polite: 明日試験があります。