Learn Russian: Это единственное место, где мы совершенно одни. - This is the only place where we are completely alone.

In Russian, the word for 'alone' (один) behaves like an adjective and must match the subject in number and gender. Because the subject is the plural pronoun мы (we), you must use the plural form одни.

Единственное and одни look like they come from the same root word. How do they differ here?

They do share the same root! However, единственное is an adjective meaning 'sole' or 'only', which modifies the neuter noun место. Одни is acting as a state of being meaning 'alone'. If you said одно место instead, it would just mean 'one place' rather than 'the only place'.

Could I use совсем instead of совершенно to mean 'completely' in this sentence?

Yes, мы совсем одни is a very natural alternative. Совершенно comes from the word for 'perfect' and sounds just a bit stronger or more absolute, but both adverbs pair extremely well with одни to mean 'completely alone'.

Why is there a comma before где? English does not use one there.

Russian punctuation rules are very strict about separating clauses. Any time you introduce a new subordinate clause (a secondary phrase with its own subject and implied verb, like мы being одни), you must separate it from the main sentence with a comma.