to bring fully to completion: to finish what was begun, or carry an appointed task all the way through
This verb is built on the ordinary word for finishing, τελέω, with a prefix that pushes it toward full, thorough completion. It is not just to end an action but to carry it all the way through to its intended result. Paul uses it for finishing what was already started: the relief collection he wants the Corinthians to complete (2 Corinthians 8:6, 8:11), the delivery of that gift he will have carried through before he travels on (Romans 15:28), and, most memorably, the good work God began and promises to finish (Philippians 1:6).
The same verb also covers carrying out a fixed, sacred duty. In Hebrews the priests keep performing the appointed services, and Moses is charged with executing the tabernacle exactly to the pattern he was shown. Here the stress falls less on ending something and more on discharging a task in full, doing everything it requires.
A reader can usually catch which shade is meant from the context, but the common thread is worth keeping in view: whether the object is a collection, a promise, a rite, or even a stretch of suffering being lived out (1 Peter 5:9), the word insists the thing is brought all the way to its finish, not merely started or attempted. It is a close cousin of τελειόω, though that word leans toward bringing something to its goal or maturity rather than simply completing it.