What is the importance and implication of Random Intercept in a mixed-model? Is it necessary to always include the random intercept in a mixed model? Can we drop it sometimes? How can we know when it is or it is not possible to drop it? Would its exclusion adversely affect the model specification? Is that effect always considerable? How can we know when it is considerable and when it is not? Are AIC and BIC the indicators of a proper model with or without the random intercept?
And also why and how is it important?
Thanks a lot.
 A: It depends on whether the mixed model is being used to adjust for temporal or clustering effects.
In a longitudinal model, a random intercept says that each subject can have its (or his or her) own starting point. Typically, random slopes are also an option, and these relate to the effect of time - each subject can have a different rate of increase. Here, dropping the intercept might make sense, it is sensible to suppose that all subjects started at the same point.
In a clustered data situation (e.g. students in classrooms) it is harder (at least for me) to see when you would drop the intercept term, but maybe I haven't had enough coffee yet. :-) 
A: The highly recommended paper by Barr et al. (2013) discusses, among others, models with random slopes but without random intercepts (p. 262). Their final recommendation clearly prefer these models to random intercept only models (p. 267):

LMEMs with maximal random slopes, but missing either random
  correlations or within-unit random intercepts, performed nearly as
  well as ‘‘fully’’ maximal LMEMs, with the exception of the case where
  p-values were determined by MCMC sampling. In addition, there was
  slight additional anticonservativity relative to the maximal model
  for the models missing within-unit random intercepts. This suggests
  that when maximal LMEMs fail to converge, dropping within-unit random
  intercepts or random correlations are both viable options for simplifying the random effects structure.

and

From the point of view of overall Type I error rate, we can rank the
  analyses for both within- and between-items designs in order of
  desirability:
  
  
*
  
*min-F' , maximal LMEMs, ‘‘near-maximal’’ LMEMs missing within-unit random intercepts or random correlations, and model
  selection LMEMs using backward selection and/or testing slopes using
  the ‘‘best path’’ algorithm.
  

Barr, D. J., Levy, R., Scheepers, C., & Tily, H. J. (2013). Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of Memory and Language, 68(3), 255–278. doi:10.1016/j.jml.2012.11.001
