To osculate is to kiss or touch with your lips. If you osculate your dog on the mouth, some of your family members will laugh while others will be disgusted.
The kind of kiss you give when you osculate is by definition either a goofy one or an almost exaggeratedly formal one, like a dapper gentleman who kisses a lady's hand. There is also a mathematical definition of osculate, which is almost like a math kiss: when two surfaces or two curves touch in at least three points. The Latin root word is osculum, kiss, or literally little mouth.