What you may be thinking of is what is called a stripper well. Usually a well that produces 10 barrels or less per day that is uneconomical to keep online at $10 or $20/barrel but becomes very economical at 80 bucks per. Maintenance on an existing well can be killer when it is required so these marginal producers sometimes get iffy.
Another thing that can be done is horizontal drilling which can also produce and area for a longer period of time but gets very, very pricey to the tune of 30 grand or more per day of drilling costs. Big, big gamble but can extend the years of profitable production for a field.
Not sure what you call them. It's a little out of my area of expertise, but a close friend of mine is a geologist and contracts to various oil companies to help them find oil. He goes all over creation doing this. In the past four or five years he said they have been investingating wells in West Texas that were long ago capped because they were thought to have played out. Many of these old wells they've looked at are producing oil again in sufficient quantaties so as to make them profitable. Of course, this is especialy true at the current price of crude, but whoever he's working for has been doing this for some time now.