Why is it called a band saw?
A bandsaw is a saw in the form of an endless balde, running over two wheels.
Bandsaws are used chiefly for cutting wood. The wheels are arranged usually one above the other. 
A bandsaw operating in "resaw" mode is a bandsaw where rough-sawn timbers of various dimensions and shapes are cut into thinner boards and/or slices.

Besides, bandsaws for woodworking are mostly small. They use the machine to replace part of the manual labor of cutting wood to improve production efficiency.

Such machines provide a stream of cutting fluid recirculated from a sump, in the same manner that a CNC machining center does. A vertical bandsaw, also called a contour saw, keeps the blade's path stationary while the workpiece is moved across it. This type of saw can be used to cut out complex shapes and angles.
 
Is the band saw used for irregular cuts?
Bandsaws are mostly used to make irregular cuts in wood, even though some kinds, like horizontal bandsaws, are more suited for heavy-duty metal cuts.