What are the most common metrics for comparing two clustering algorithms (especially density based clustering) When it comes to compare a new clustering algorithm, one always wants to show the advantages of his/her method over existing and well known methods. Going this way may mislead one to ignore disadvantages proposed method.
For clustering results, usually people compare different methods over a set of datasets which readers can see the clusters with their own eyes, and get the differences between different methods results.
There are some metrics, like Homogeneity, Completeness, Adjusted Rand Index, Adjusted Mutual Information, and V-Measure. To compute these metrics, one needs to know the true labels of data-set, so we may test algorithms with classification data-sets to have true labels and then evaluate results.
Another metrics, like Silhouette Coefficient works only with data and clustering results.
I want to know what measures are most preferred and if there is any other metric which does not require true labels of data-set. 
 A: It seems that you ask for internal validation measures (using only the data without reference to anything else) to compare different algorithms. This is fraught with perils. See e.g. the answer by Anony-Mousse here: https://stats.stackexchange.com/questions/88550/using-the-gap-statistic-to-compare-algorithm. My original answer (below) discusses external validation measures. It is still useful perhaps.
Here are two standard approaches (there may be more). The first is to use a gold standard and compute a distance or similarity between clusterings. Many people use the adjusted Rand index, but I think the Variance of Information distance (VI) or split/join distance are better suited for this. (See e.g. my answers here: Comparing clusterings: Rand Index vs Variation of Information and here: Forgiving measure for external cluster validation). This is still not straightforward -- a clustering can be consistent with a a gold standard (be a sub-clustering or super-clustering) and this has to be taken into account but often is not. I've seen a case where a very coarse classification was used (large classes), and the other clustering algorithms just produced more fine-grained result (subclusterings with respect to this coarse gold standard).
The second is to have some kind of annotation -- a classification associated with the nodes, where each node can have multiple labels. In biology this could be the GO (Gene Ontology) classification. It is then possible to compute an enrichment score (e.g. using the hypergeometric distribution) for each cluster (check which labels are overrepresented in the clustering). This is also not entirely straightforward (as collections of P-values have to be compared), but both approaches are definitely very informative if care is taken.  
