Random intercepts

Random intercepts are

the way the model adjusts the average (fixed) intercept to better match the data from each level of the grouping variable(s).
one number per level of the grouping variable, which is added on to the fixed intercept to produce the group-specific intercept.
also called “varying intercepts” or “intercept adjustments”.
represented by the 1 in the random effect lme4 syntax: (1 + p | g), where g is the grouping variable. If a model has only random intercepts by g and no random slopes over p, the random effect term is (1 | g).

To work out whether your model should contain random intercepts, see Identify possible random effects.

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