When you're entrenched, you're dug in. Sometimes that means you're literally in a trench, but usually it means you just won't budge from a position or belief.
Entrenched things are buried so solidly that they can't move — or just behave like they're firmly lodged in some deep hole. Usually this word refers to views people hold very strongly. Having turkey on Thanksgiving is a tradition that's entrenched in American culture — it's long been established and isn't going anywhere. When you're entrenched, you're being stubborn or consistent, depending on your view.