When using だろう (the conjugated auxiliary verb) is grouped with か (the question particle), it presents the statement before it as something that the speaker is unsure about. In this way, it sounds similar to ‘I wonder if (A)’ in English.
To use this construction, the statement before だろうか must be nominalized (turned into a noun-phrase). This is done through the use of の, or なの (in the case of な-Adjectives and nouns.

• 私（わたし）たちのチームは勝（か）てるのだろうか。
  I wonder if our team can win.
• 彼女（かのじょ）は本当（ほんとう）に彼氏（かれし）が欲（ほ）しいのだろうか。
  I wonder if she actually wants a boyfriend.
• あの人（ひと）は大丈夫（だいじょうぶ）なのだろうか。
  I wonder if that person is okay.
• あれは本当（ほんとう）に食（た）べ物（もの）なのだろうか。
  I wonder if that is really food.

Despite the grammatical rule relating to nominalization using の (or なの), this part of the phrase is often omitted.

• トムは私（わたし）を誘（さそ）うだろうか。
  I wonder if Tom will invite me. (の omitted)
• あの人（ひと）は本当（ほんとう）に弁護士（べんごし）だろうか。
  I wonder if that person is actually a lawyer. (なの omitted)

だろうか is quite masculine, so in order to sound more polite/ladylike, でしょうか may also be used instead.

• そんな高（たか）い鉛筆（えんぴつ）をトムが買（か）うのでしょうか。
  I wonder if Tom will buy a pencil that expensive.
• あの車（くるま）はキムの車（くるま）なのでしょうか。
  I wonder if that car belongs to Kim.