How to dig through an aquifer
An aquifer is one or more layers of rocks that spew water in all four horizontal directions, as well as down, if there is empty space there.
Aquifers are discussed in depth here: Aquifers
There are seven or eight methods of dealing with aquifers discussed there.
In addition, the Double Slit method is one of the more straightforward, though painstaking, methods for dealing with any aquifer: Double Slit Method
The Double Slit method uses pumps to pump water from one side of your work site, draining excess water into other aquifer blocks, while miners dig out the walls around the work site, and masons or carpenters then wall off those dug out sections. After each layer of the aquifer is secured, the next lower layer is probed, and, if there's another layer of the aquifer, the same process is repeated. Once the bottom of the aquifer layers has been found, a slightly different draining procedure is used to secure the lowest aquifer layer.
An aquifer is a layer of water-bearing stone found in certain biomes. Digging the stone in question is possible but any removed stone will be replaced with water. In addition, exposed stone in that layer will slowly produce more water until you smooth it.
Sometimes an aquifer covers an entire layer of a site. Other times, if the site spans two biomes, you may be able to dig around it.
The simplest way to dig through the aquifer is to use screw pumps to remove the water as you dig downwards and have your dwarfs smooth any walls which are left exposed.
For more complex methods, the wiki has an extremely good page on aquifers and dealing with them.