The definition of liaison is the illicit meeting between two lovers. The reason why the book is called Les Liaison Dengereuse (or in English "Dangerous Liaisons") is because the young lovers, Cecile and Danceny, are going against the will of their superiors, which is that Cecile is to marry Gercault. Also the two corrupt aristocrats, Merteuil and Valmont, are attempting to destroy their love.
The characters, Merteuil and Valmont are the evil rich aristocrats who are motivated by cruelty, power, and jealousy, they exist only to destroy other's love. Their relationship is the most dangerous of the play as they are two evils on the same side, and as such can turn on each other at any moment. Also we get indication in the play that Merteuil has some control over Valmont and as such can make him do her evil plots to destroy people's names. For example, at the start Merteuil wanted Valmont to exact revenge on Gercault, and managed to persuade him with not too much difficulty.
In the middle of the play Valmont seduces Cecile, this a dangerous liaison as she is at that time in love with Danceny. This is helped by Merteuil influence on her which is that of trust, as they often talk to each other during operas, for example even after Valmont has successfully had sex with her, she expresses trust in Merteuil and confides in her this information and asks her advice, "He has a way of putting things..." Valmont seduces her by gaining her trust by delivering her letters from and to Danceny, and telling her that it will aid their love if she were to have "experience in the matter".
Merteuil later in the play has a relation to Tourvel, her motives are mostly that of revenge...