What it the straw size soft vein at the back of the bottom of each testicle? it tenders when touched and some time swells after peeing. what is it?

Your vas?: I'm betting you discovered your vas deferens. One on each side. It is a little bit tender. It's firm and about as big around as a coffee straw. The epididymis is bulkier, not quite so tender, and softer. It's good to get acquainted with your equipment, especially checking every few weeks for a testicular lump -- cancers aren't that common but are very much worth detecting eraly.