Does this picture show sister chromatids or homologous chromosomes? If they are homologous then what is YY? If they are sister chromatids then do homologous chromosomes ever appear like this (with the centromere)? Do sister chromatids exist only in prophase-I to anaphase-I of meiosis?
Since I have used more than 1 image in my answer; with numbers starting from 1; I'll call your provided figure as figure-0
What is shown in following picture?
Though the image showing many things; in overall it is an image of a set of chromosomes; seemingly almost certainly from human. The chromosomes has been stained with a banding method (though I'm not sure about which banding method used).
Are they sister chromatids?
Could not be answered in few words. The image contains sister chromatids. But the entire image in OP (fig-0) could not be described as "image of sister chromatids".
All the chromosomes in this image are in metaphase. so each chromosome is made up of 1-pair of chromatids which are sister-chromatids to each-other.
Fig 1. 
This total image showing 1 chromosome (on or before metaphase) showing 2 chromatids (written as A and B). A and B are sister chromatids because their DNA-content resulted from same mother-DNA molecule when replication happened at S-phase of cell-cycle; so their DNA content is basically same.
or homologous chromosome?
No. They are Not homologous. However Chromosome X and Y do have some homologous portion (Pseudo-autosomal regions). Question-figure (Fig-0) neither shows a complete set (46 chromosomes) from 1 diploid cell; nor shows its half (the haploid set) (that is causing your confusion about X and Y; let me proceed...) . Fig-0 shows each-type of human chromosome in 1 piece.
If they are homologous then what is YY?
YY karyotype does not normally exist in human.
Karyotype of human male:
2 x (22 autosome) + 1 X-chromosome + 1 Y-chromosome.
Karyotype of human female
2 x (22 autosome) + 2 x X-chromosome.
Then how are homologous chromosomes?
Fig-2 (photo Taken from iGenetics; by peter J russell; Ed-3 ebook). Slightly modified.
it shows a complete set of chromosome from a human male cell. Note that it doesn't contain YY but only a single X plus Single Y.
And from same photo; what are sister and nonsister chromatids, and homologous and heterologous (non-homologous) chromosomes.
Fig-3 .
Reference: iGENETICS by Peter J. Russell, Ed-3.
The picture can be a bit misleading because it represents 22 autosomes (autosome = non-sexual chromosome) while there are 22 pairs of autosomes (so the homologous chromosome is not represented). And it represents the whole pair of sexual chromosomes (the individual is a male).
homologous chromosome
For each autosome represented, there is another one which would be its homologous chromosome.
Sister chromatids
As you can see, each chromosome looks like an X. The right and left side of this X there are chromatids (two chromatids per chromosome here). The chromatids are attachedf by the centromere.
Note
Please note that 1) chromosomes looks like they are represented here only during the metaphase (when they are condensed). 2) A chromosome is not always made of two chromatids. Sometimes it is made of single chromatid only. It depends on the moment in the cell life cycle.
