There's a couple types of springs. There's the classic "water bubbling up from the ground" type called an Artesian spring; since that's likely what you're asking about I'll stick with that.

Water flows through the ground just like it does on the surface, but slower. It flows down slopes and pools at deep locations. Groundwater is slower because of all the stuff like sand, gravel, clay, boulders, etc that get in the way. The water is under pressure (like a soda sorta) much of the time because of all the heavy ground on top of it compressing things.

In hydrology there's a concept called a Potentiometric Surface; not quite the same as a water table. The water table is simply the depth that the top of the water is at (e.g. 10m below surface). The Potentiometric Surface is the height that that water would rise to if the pressure on it was removed (like if there were a bunch of wells). This is what drives artesian wells/springs to bubble up from the ground. In those places, the Potentiometric Surface is actually above the ground level, and so the water, when released from its pressure by a well, bubbles up until it can't anymore (gravity and fluid dynamics etc pull the water down again).