Example 1: French defense: Qb6
A queen on b6 puts pressure both on the b2 pawn (sometimes threatening to take when the pawn becomes weak after the bishop develops) and more importantly, the central d4 pawn.
1. e4 e6 2. d4 d5 3. e5 c5 4. c3 Nc6 5. Nf3 Qb6
Putting pressure on White's center is a main strategy of the French and Caro-Kann advance variations, and Qb6 an important move to that end.
In the Poisoned Pawn Variation of the Najdorf Sicilian, Black often takes the "poisoned" b2 pawn. This also happens in other openings, e.g. a line in the Caro-Kann Advance, Short variation. When you take on b2, you need to make sure the queen does not get trapped!
Example 2: Caro-Kann defense: Qa6
1. e4 c6 2. d4 d5 3. e5 Bf5 4. h4 h5 5. Bg5 Qb6 6. Bd3 Bxd3 7. Qxd3 Qa6
In many lines of the Caro-Kann defense (and the French defense), the queen gets developed to b6 in order to put pressure on the White center. In this example however, I'd like to show you the move Qa6, which is also a very common theme (sometimes the queen also comes from a5 after giving a check). The goal of Qa6 is to trade off the queens.
Example 3: Qa5+ tactics
Qa5+ is a common tactic/resource, often picking up something on the 5th rank. There are countless examples, take this one as an illustration. The move Qa5 is also common in a lot of other openings (e.g. Grünfeld defense).
1. e4 c5 2. Nf3 e6 3. d4 cxd4 4. Nxd4 Nf6 5. e5?? Qa5+
Example 4: Taimanov Sicilian: Qc7
1. e4 c5 2. Nf3 e6 3. d4 cxd4 4. Nxd4 Nc6 5. Nc3 Qc7
Here, the queen gets developed early to c7. One idea of the move is to control the center (e5).
While I hope that these examples show, how the queen is used in the opening, I strongly encourage you to remember that generally, you shouldn't use your queen too early in the game.
EDIT: Of course, the same ideas apply with colours reversed, e.g. Qb3 or Qc7 in many Queen's Pawn openings. There is also Qh5 when Black has not castled, attacking on the h5-e8 diagonal (if the f-pawn has moved, with check - this is a reason why moving the f-pawn in the opening is very often a bad idea).