Beta values for mixed models

Mixed models are so-called because they estimate both random effects and fixed effects.

There is no universal acceptance of what constitutes a random effect and a fixed effect, but a good starting point is to assume that a fixed effect is a variable that remains constant across individuals or experimental units, for example, the "effect" of being male or female, or the "effect" of a treatment in a trial or experiment, whereas a random effect is something that varies across individuals, such as the "effect" of receiving a treatment at a certain hospital, or the "effect" of being a particular individual when repeated measurements are taken. Note that when a variable is specified as a random effect, it usually also has a fixed effect.

Without any psychic ability to read your professor's mind, I suspect they may be asking to know what the random effects and fixed effects from your model(s) are. Taking the output from lmer() using the built-in sleepstudy dataset:

> require(lme4)
> m0 <- lmer(Reaction ~ 1 + Days + (1 + Days|Subject), data=sleepstudy)
> summary(m0)

which produces:

Linear mixed model fit by REML ['lmerMod']
Formula: Reaction ~ 1 + Days + (1 + Days | Subject)
   Data: sleepstudy

REML criterion at convergence: 1743.6

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.9536 -0.4634  0.0231  0.4634  5.1793 

Random effects:
 Groups   Name        Variance Std.Dev. Corr
 Subject  (Intercept) 612.09   24.740       
          Days         35.07    5.922   0.07
 Residual             654.94   25.592       
Number of obs: 180, groups:  Subject, 18

Fixed effects:
            Estimate Std. Error t value
(Intercept)  251.405      6.825   36.84
Days          10.467      1.546    6.77

Correlation of Fixed Effects:
     (Intr)
Days -0.138

This shows the random effects and fixed effects estimates and (for the fixed effects their standard errors and t values). Other packages such as nlme will produce equivalent output. Here we have a random intercept for Subject because observations are clustered (repeated) within subjects, so each subject has it's own intercept; and a random coefficient for Days so that each subject has its own slope for the variable Days.

As for "beta values", these are usually the estimates of the fixed effects, in the above example, we have 2 fixed effects, the intercept (often called "beta-0" or $beta_0$, estimated here as 251.405) and the fixed effect of Days (often called "beta-1" or $beta_1$ because it is the first fixed effect after the intercept, here estimated as 10.467).