Laughter appears to be modeled by the mother during the first year and stabilizes in the infant by the second year. (Martin, Rod A. White Papers. Do Children Laugh Much More Often than Adults Do?)
Laughter in infants occurs about a month after smiling. May be observed as early as 5-9 weeks of age. By 4 months it is firmly in place. (Rose, Kenneth Jon. The Body in Time. NY: John Wiley & Sons, Inc., 1988, pp 150-152)
Studies: babies under age 1 were 15 times more like to laugh if tickled by their mothers than by strangers. Babies must feel safe in order to laugh. (Dossey, Larry, MD. Healing Beyond the Body. Boston: Shambhala Publications, Inc., 2001, pp 133-149)
Laughter emerges early in human development, being reliably elicited through tickling by about 4 months of age (Sroufe & Waters, 1976). Children born both deaf and blind also laugh at roughly the same age (Eibl-Eibesfeldt, 1989), indicating that this signal is deeply rooted in human biology (Deacon, 1989). (Bachorowsk, Jo-Anne I, PhD, and Michael J. Owren, PhD. Laughing Matters. Psychological Science Agenda, Volume 18: No. 9, September 2004.)
By about age 12 months a child begins to laugh at unusual or inappropriate adult behavior (e.g., funny faces, walking on all fours). By age two they begin to create their own juxtapositions and laugh at incongruities. (Branson, Roy, PhD. The Sacredness of Laughter. WA: Spectrum, Vol. 26, No. 4, January 1998, pp 45-46)
Louis Franzini, PhD, author of Kids Who Laugh: How to Develop Your Child's Sense of Humor: laughter is critical to a child’s development. It plays an important role in stress diffusion, developing self-esteem, learning to problem solve, and honing social skills. (Abedon, Emily Pearlman. Why Laughter Is a Sign of Learning.)
Refer to Laughter – Humor and the Brain for additional information.