In the oil field, when a vertical cylinder coring sample is taken, how is the core severed at the bottom and removed from the formation?
Coring is an important method to get detailed information about the formations, however, it's very time-consuming and costive, so it's only applied when geologists and reservoir engineers need high-resolution data. It is more common in ore prospecting and mining, mapping for infrastructure and basic geological understanding of the 3-dimentional structures. Lake sediment cores are used for varve chronology. Marine sediments are cored to understand the oceanographic development, date events as slope processes and changes in glaciation and climate. Ice is cored to measure variations in the atmosphere to understand past climate and vulcanism. 

To sum up, coring is a very important tool for geologists, not only in oil fields. Cheaper, and faster, alternatives to coring is mud logging, a very common practise when drilling in oil fields. 

So how to get the cores up from the borehole?

There are some different systems used, depending on the rock/sediment and investment involved. 

For deep drillings in hard formations, a core catcher is used in a two-barrel system. The corehead (the drill bit) is connected to the outer barrel, while the core is pushed into an inner barrel. The lower part of the inner barrel is called the shoe just above the drill. In the shoe, there is a core catcher that wedges between the barrel and the core. The core catcher is a split ring with a rough inside that holds the core. When you pull the core, it snaps off as the lower and upper shoe squeezes the catcher. The core catcher works somehow like an expansion anchor, but inverted, grabbing the core on the inside of the barrel rather than the wall on the outside. The inner barrel also protects the core from the rotating outer barrel. 

For softer sediments, the core catcher is made out of springs that close inwards as the core is pulled. It is sometimes called basket retainer and can be in metal or plastic. 

For smaller gravity corer, there is a valve on top of the tube that is closed and the underpressure in the tube holds the core in place. Sometimes... 

Check the linked pictures, they probably explain better how it works, but I couldn't find any good CC.