This is a bid of a suit with no intention that the bid suit should be the trump suit. The term covers several quite different situations.
1. In a slam investigation, when the partnership has agreed a trump suit (either explicitly or implicitly), the cue bid of a suit shows a control in that suit. By agreement this may be first-round control (an ace or void), or either first- or second-round control. To avoid confusion with the other types of cue bid, some call this a control-showing cue bid.
After East’s 3♠ raise sets spades as trumps, West’s 4♣ and East’s 4♦ are both (control-showing) cue bids. 4♣ must show interest in a slam because, with no such interest, West would simply raise 3♠ to 4♠.
2. In a contested auction, a bid of the opponents’ suit is called a cue bid: it may be a general forcing bid (when no suitable alternative is available) or have a conventional meaning.
Popular nowadays is for East’s 3♣ cue bid in the first sequence to show a value raise to at least 3♥. The older treatment is to play it as asking for either a full or half a club stopper.
In the second sequence, where partner’s action is an overcall, it is almost universal to play the 3♣ cue bid as showing a sound (heart) raise, also known as an Unassuming Cue Bid.
3. A direct overcall in the suit an opponent has opened conveys the message of a very powerful hand or, more popularly, some form of two- suited hand.
The popular Michaels convention and the rarer Ghestem convention both use this type of cue bid to show a two-suited hand.