A PCB via is a hole that connects a signal between two surfaces - the hole is typically very tiny and chemically plated to be conductive - they are essentially wires that carry signals between layers.
Because vias are typically small, they can't carry much current, so if you need heavier currents you simply put a whole load of vias together, 10, 20, whatever you need. Layout tools generally include calculators that let you know the current carrying capacity of different types and sizes of vias in differing environmental conditions.
In the simplest case of a double sided PCB, all vias will go from one side to the other (we'd say "top to bottom"), but in the case of a 4 layer PCB having two internal layers, a via could go from 1-2,1-3,1-4,2-3,2-4 or 3-4.
A via that is completely internal, e.g 2-3 in the 4 layer example, is not visible from the top or bottom and is thus referred to as a "buried via". A via that goes from an external to an internal layer, e.g. 1-3, is known as a "blind via".
PCBs are actually quite complex things to get right. At really high frequencies, e.g. for. PC motherboards that run in the GHz range, the length of a signal trace is critical as, believe it or not, signal propagation is close to the speed of light so tiny inconsistencies in trace lengths can cause horrendously difficult timing issues, i.e. bugs. We're talking timing sensitivity in the nS range. Industrial PCB board layout programs will actually make sure that traces that are timing sensitive are exactly the length they need to be by inserting tiny wiggles and using other tricks. This obviously gets more complex where signals move between layers too, as via length also has to factored in. With fast digital signalling, the divide between the digital and analogue domains becomes blurred - digital signals are just a special case of analogue. The traces on the PCB then start acting as transmission lines, and that's another world again.
PCB layout is mainly science but has a lot of art involved. It's a highly skilled job and although most electronic engineers can maybe layout a simple PCB that works fine, a specialist at work is a joy to behold.
Commercially, layout is a distinct bit of the DFM (design for manufacturing) process.