It is a type of Smilax with several common names. Bamboo vine, Greenbriar, Catbriar. You won't be able to pull it up unless it's very young. Established plants have a huge, tough, knotty root. Unless you want to use chemical means it will have to be dug out and it will be a job. The good news is that it doesn't have a tap root. The root usually grows a foot or so underground and spreads out. It looks a little like a bunch of co-joined, jointed, potatoes but is very tough. If you don't get it all it will sprout again. Good luck.

HAH! That is my old friend the "Klingon Vine". We lived in a 100 year old farmhouse and the front fence was covered with this vine. Just had to let it be and keep it cut back, no doing anything with it. We are big Star Trek fans so it was natural to name it after the tough and "thorny" Klingons. That is what I will always call it--it should be their national plant!!!!!!!
Betsy