Genes-of-interest analysis between organisms
In RNA sequencing, total RNA is extracted, then mRNAs are purified out from the sample using polyT columns (since mRNAs have a polyA tail, this will attach to a polyT DNA chunk that is attached to some solid surface). This step is necessary because ribosomal RNAs are much more abundant than mRNAs, and for sequencing you only want to use mRNAs. Then the purified mRNA is transcribed to DNA using reverse-transcriptase and finally the cDNA is sequenced.

As for microarray a set of single stranded oligonucleotids (short DNA sequences) (aka. probes) are attached to a solid (usually glass) surface as small dots. This the so called DNA/RNA-chip. Then the chip is washed with the fluorescently labelled singlestranded sample, and the complementary strands attach to each other, the rest is washed away - this is a so called hybridization technique. The signal is then detected by reading the fluorescence signals at each spot.

The microarray technique is quite old and is less sensible and the number of probes are limited / array, while with RNA sequencing you can sequence the total mRNA population of each organism and examine anything you want. Also since most sequencing techniques include a PCR step less starting material is sufficient, and has higher sensitivity and specificity.

These wiki pages on RNA-seq and microarray technique provide detailed info on both.