I am new to learning about RNAseq analysis and am confused as to what exactly sequencing depth refers to. For example, if I need to calculate "how deep each sample was sequenced" does this refer to the total number of paired end reads that came out of the sequencer OR the number of paired end reads that mapped?
Depth is commonly a term used for genome or exome sequencing and means the number of reads covering each position. But that is for RNA-seq totally pointless since the coverage pattern is so uneven due to differences in expression.
More commonly, in RNA-seq the term "number of reads" is used, for example,10 million reads or 100 million reads. If all goes well a high percentage of the reads should align, ~90%. That makes the number of reads out of the sequencer not so different from the number of mapped reads. I would report the number of reads sequenced unless it's very different from what is aligned. But then you should also figure out why it's such a big difference.
I want to give you a very intuitive but maybe not very accurate explanation: You can imagine each base (A/G/C/T) is a grain of rice. Suppose you sequenced a big of rice (say 100M bases), and then fill them into a very narrow but very long rice box with width = 1 grain of rice. The rice height in the box would be the sequence depth. For WGS, the length would be length of whole genome. For RNAseq, people might only consider reads mapped onto exons, and also only consider the length as union of exon length.
"how deep each sample was sequenced" means "how many reads (or pairs)" were sequenced for each sample. Some results will vary with the number of sequenced reads (the sequencing depth), therefore it is an important information.