Variance-covariance matrix in lmer I know that one of the advantages of mixed models is that they allow to specify variance-covariance matrix for the data (compound symmetry, autoregressive, unstructured, etc.) However, lmer function in R does not allow for easy specification of this matrix. Does anyone know what structure lmer uses by default and why there is no way to easily specify it?
 A: The FlexLamba branch of lmer provides such a functionality. 
See https://github.com/lme4/lme4/issues/224 for examples how to implement a specific structure of errors or random effects. 
A: To my knowledge lmer is not having an "easy" way to address this. Also given that in most cases lmer makes heavy use of sparse matrices for Cholesky factorization I would find it unlikely that it allows for totally unstructured VCV's. 
To your address your question on "default structure": there is not a concept of default; depending on how you define your structure, you use that structure. Eg. using random effects like : $(1|RandEff_1)+(1|RandEff_2)$ where each random effect has 3 levels will result in unnested and independent random effects and a diagonal random effects VCV matrix of the form:
$R = \begin{bmatrix} 
\sigma_{RE1}^2 &  0& 0 & & 0 & 0  & 0\\   
0 &  \sigma_{RE1}^2& 0 & & 0 & 0  & 0\\ 
0 &  0& \sigma_{RE1}^2 & & 0 & 0  & 0\\ 
0&  0& 0 & &  \sigma_{RE2}^2 & 0 & 0 \\
0 &  0& 0 & & 0 &  \sigma_{RE2}^2 & 0\\
0&  0& 0 & &  0& 0 & \sigma_{RE2}^2  \\\end{bmatrix}$
All is not lost with LME's though:
You can specify these VCV matrix attributes "easily" is you are using the R-package MCMCglmm. Look at the CourseNotes.pdf, p.70. In that page it does give some analogues on how lme4 random effects structure would be defined but as you'll see yourself, lmer is less flexible than MCMCglmm in this matter.
Half-way there is problem nlme's lme corStruct classes, eg. corCompSymm, corAR1, etc. etc. Fabian's response in this tread gives some more concise examples for lme4-based VCV specification but as mentioned before they are not as explictly as those in MCMCglmm or nlme.
A: Mixed models are (generalized versions of) variance components models. You write down the fixed effects part, add error terms that may be common for some groups of observations, add link function if needed, and put this into a likelihood maximizer.
The various variance structures you are describing, however, are the working correlation models for the generalized estimating equations, which trade off some of the flexibility of the mixed/multilevel models for robustness of inference. With GEEs, you are only interested in conducting inference on the fixed part, and you are OK with not estimating the variance components, as you would in a mixed model. For these fixed effects, you get a robust/sandwich estimate that is appropriate even when your correlation structure is misspecfieid. Inference for the mixed model will break down if the model is misspecified, though.
So while having a lot in common (a multilevel structure and ability to address residual correlations), mixed models and GEEs are still somewhat distinct procedures. The R package that deals with GEEs is appropriately called gee, and in the list of possible values of corstr option you will find the structures you mentioned. 
From the point of view of GEEs, lmer works with exchangeable correlations... at least when the model has two levels of hierarchy, and only random intercepts are specified.
