Oboje su kandidati.

Here oboje means both or both of them. It refers to two people together and acts as the subject of the sentence.

So Oboje su kandidati. = Both are candidates.

Because oboje refers to two people, so the verb agrees with that plural meaning.

• su = are
• je = is

Even though oboje may look like a singular/neuter form to a learner, its meaning is plural here, so su is correct.

Because kandidati is the predicate noun after su, so it is in the nominative plural.

Compare:

• Oboje su kandidati. = Both are candidates.
• oba kandidata = both candidates

In oba kandidata, the numeral directly modifies the noun, which gives a different form. But in Oboje su kandidati, you have a full sentence with are, so kandidati is nominative plural.

These forms are similar, but they are not used in exactly the same situations.

• oboje: both people together, often a mixed-gender pair, or when gender is not important
• obojica: both men
• obe: both women / feminine nouns
• oba: both masculine or neuter nouns, often used directly before a noun

For people, Serbian often prefers:

• Obojica su kandidati. = both men are candidates
• Obe su kandidatkinje. = both women are female candidates
• Oboje su kandidati. = both people are candidates, often a mixed pair or unspecified

Not always, but that is a very common use.

It often refers to:

• a male-female pair
• two children
• two people whose gender is not being emphasized

So in many contexts, oboje strongly suggests both of them together, especially as a pair, rather than focusing on their individual genders.

Because in Serbian, the masculine plural is normally used for mixed groups.

So if the two people are, for example, one man and one woman, kandidati is the normal plural form.

If both were women and you wanted to say that explicitly, you would more naturally say:

• Obe su kandidatkinje.

You can, but it is usually unnecessary.

Oboje already does the job of the subject, so:

• Oboje su kandidati. is the normal, natural sentence.

Adding oni can sound more emphatic or contrastive in some contexts, but most of the time Serbian simply says oboje.

Yes. Serbian word order is fairly flexible, though Oboje su kandidati. is the most neutral order.

Other orders are possible for emphasis, for example:

• Kandidati su oboje.

That still means basically the same thing, but it can sound more marked or emphatic depending on context.

You would normally use a different form:

• Obojica su kandidati. = both are male candidates / both men are candidates
• Obe su kandidatkinje. = both are female candidates

So oboje is not the only way to say both; the exact choice depends on who the two people are.

You can think of it as a both-type word that behaves somewhat like a pronoun here.

In this sentence, it stands on its own, without a noun after it, and means both of them. In grammar books, it is often treated as part of the system of collective numeral forms, but for a learner the most useful thing is this:

• here it functions like a subject meaning both of them.