These numbers tell you the size of the yarn and how many plies it has. It's called the "count system".
The first number is the size - the larger the number, the finer the yarn. (The physical sizes can vary for different fiber types, but they always follow that rule.)
The second number is the ply, how many singles make up the yarn.
So both of your examples are six-ply yarns, but the 50/6 yarn will be finer than the 34/6.
If you spin yarn, these numbers will tell you how much fiber you need to make a certain length. Fibers have a count associated with them (also different for each fiber type - for instance, cotton is 840 but linen is 300). That number is how long a single ply of yarn would be, spun at size 1, if you used 1 lb of fiber. So 1 lb of a 1/1 cotton would be 840 yds long. However, if you wanted to make 10/2 cotton, you would multiply by the fraction:
(10 / 2) * 840 or 4200 yds of yarn, from the same weight. This value (yards per pound) is called the "grist", and is directly correlated with the count system number.
Well, I know a heck of a lot about yarn and gauge, but I have never seen numbers like 50/6 or 34/6. Can you post a picture of where you see these numbers?
The square in the middle is telling you that using US size 8 knitting needles (5 mm needles), a 4x4 inch (10x10 cm) square of material should contain 20 stitches and 23 rows if knit in stockinette stitch. Or, for crocheting, if you use a size I-9 hook (5.5 mm) the same 4x4 inch square should include 16 single crochets in length and 18 rows of single crochet in height.
The format of numbers that you are showing does not look like the way yarn makers designate sizes for yarn made specifically for knitting/crochet. Instead the format of one number/another number looks like the way sizes are designated for yarn designed to be used for weaving (which is generally much finer that what is used for hand knitting/crochet). Even so, as I said, those numbers don't make alot of sense to me.
Here is a chart that shows how many strands of yarn you should use across the width of a weaving loom for a close-spaced/medium-spaced/widely-spaced woven pieces. Notice the column down the left hand side shows many sizes in the same format that you mention in your question (although none of them match the numbers you use).
I couldn't easily find any images that show a comparison of sizes of weaving yarn and knitting yarn, nor any that show the yarn with a coin or something so you can get a sense of the actual width, but here are a couple of pictures for you:
Even with the pictures, though, it is hard to understand the actual size of the yarn if you were looking at it--and it can be tough to find any references that make the comparison for you. However, I was able to find a post in this thread (scroll down to the posting by Sally Orgren) that states
Some folks think 20/2 looks like sewing thread.
Based on that, you can see that the weaving yarns tend to be pretty fine.