Interpreting random effects lmer

General
#1

I'm having some trouble interpreting the "random effects"-section in a summary generated from an lmer-model. I want to decide if my random effects are significant based on the output of summary, but I am not sure how to go about this. I have added my model and the summary()-output below.

Model
FertilizerModel <- lmer(root ~ fertilizer + week + fertilizer:week +(1+ week|plant:fertilizer))

Output summary(FertilizerModel)

Random effects:
 Groups           Name        Variance  Std.Dev. Corr
 plant:fertilizer (Intercept) 0.0306727 0.17514      
                  week        0.0007169 0.02677  1.00
 Residual                     0.2171683 0.46601      
Number of obs: 60, groups:  plant:fertilizer, 12
...
#2

A random effect is not what most of those of us who are not statisticians would expect.

Parameters associated with the particular levels of a covariate are sometimes called the “effects” of the levels. If the set of possible levels of the covariate is fixed and reproducible we model the covariate using fixed-effects parameters. If the levels that we observed represent a random sample from the set of all possible levels we incorporate random effects in the model.

There are two things to notice about this distinction between fixed-effects
parameters and random effects. First, the names are misleading because the distinction between fixed and random is more a property of the levels of the categorical covariate than a property of the effects associated with them. Secondly, we distinguish between “fixed-effects parameters”, which are indeed parameters in the statistical model, and “random effects”, which, strictly speaking, are not parameters. As we will see shortly, random effects are unobserved random variables.

Taken from this tutorial.