I am using the Tartine starter.
I dump 80% and re-feed water and 50/50 flour.
Assuming you've been feeding it 50/50 consistently, you have a starter hydration of 100%.
How do I know what percentage I have been feeding it?
I dump out about 75% to 85% and mix a little water and 2 finger scoops of flour back in.......
Until you measure your feeding routine, I don't think you CAN know what percentage your starter is. If you're going to be doing any good amount of baking, you may as well buy a scale anyway, at that point knowing your starter's hydration will become trivial.
I misread/interpreted what you said. I thought you said you were feeding it 50% water and 50% flour. Without a scale, there is no reliable way to determine what the hydration of your starter is. It's worth the $35 to buy a digital scale. If you're at all serious about baking, you need to weigh all ingredients and abandon cups and tbls, dashes of this, etc. On this website, you'll find that just about every recipe (formula) is expressed in units of "bakers percentage" which is all based on weight.
So my next question is; What is the difference between the Tartine starter and let's say a 50% hydration starter or a 65% hydration starter?
you'll probably find it helpful to learn "Baker's Math." If you type this phrase in to the search bar on this site, you'll find many good results that walk you through the calculations. It'll serve you well in all your future baking endeavors, and it'll help you figure out how to calculate hydration. To answer your question, a Tartine starter is 100% hydration (equal amounts of water and flour by weight). A 50% hydration starter is much stiffer than the batter-like tartine starter, as it contains only half as much water as flour (again, by weight). The 65% hydration starter will also have the consistency of a kneadable dough, but will be less stiff than a 50% hydration starter.
by dividing the weight of the water by the weight of the flour and multiplying by 100 to get the %.
So... a 50% starter when weighed would have twice as much flour weight as water weight. For example... 50 grams of water to 100 grams of flour which is a nice dough (50/100= 0.5 and x 100= 50%) not too soft or firm with AP (all purpose) wheat flour. A 65% hydration the dough like starter would feel softer and slouch more. At 75% even more like a shapeless blob. If using whole wheat flour or bread flour, the flour absorbs more water so the starter will be firmer than with AP.
A 100% AP starter is more like a thick batter. The water weight and the flour weight are the same but the volumes are very different; the flour volume will be about twice that of the water. So you can see that as hydration goes up, the starter can go from being dough like to soup and even on to milk like at 200%. As the starter matures, it will get thinner or more fluid.
if you spent all of 10 minutes using the search box, or the main page how to guides you would find the answer to all of the rather basic questions. And further advancement in every respect after mastering the basics. Every conceivable subject for a new baker has multiple postings on how to maintain starters, bakers percentage, and many guides on how to start out and then advance. Spend a bit of time and you will know more than you bargained for. Teach a man to fish or give him a fish, which is the better propostion? I mean this in a positive sense, not meaning to be critical.