Host-side drivers for USB devices talk to the "usbcore" APIs. There are two. One is intended for general-purpose drivers (exposed through driver frameworks), and the other is for drivers that are part of the core. Such core drivers include the hub driver (which manages trees of USB devices) and several different kinds of host controller drivers, which control individual busses.
The device model seen by USB drivers is relatively complex.
USB supports four kinds of data transfers (control, bulk, interrupt, and isochronous). Two of them (control and bulk) use bandwidth as it's available, while the other two (interrupt and isochronous) are scheduled to provide guaranteed bandwidth.
The device description model includes one or more "configurations" per device, only one of which is active at a time. Devices that are capable of high-speed operation must also support full-speed configurations, along with a way to ask about the "other speed" configurations which might be used.
Configurations have one or more "interfaces", each of which may have "alternate settings". Interfaces may be standardized by USB "Class" specifications, or may be specific to a vendor or device.
USB device drivers actually bind to interfaces, not devices. Think of them as "interface drivers", though you may not see many devices where the distinction is important. Most USB devices are simple, with only one configuration, one interface, and one alternate setting.
Interfaces have one or more "endpoints", each of which supports one type and direction of data transfer such as "bulk out" or "interrupt in". The entire configuration may have up to sixteen endpoints in each direction, allocated as needed among all the interfaces.
Data transfer on USB is packetized; each endpoint has a maximum packet size. Drivers must often be aware of conventions such as flagging the end of bulk transfers using "short" (including zero length) packets.
The Linux USB API supports synchronous calls for control and bulk messages. It also supports asynchronous calls for all kinds of data transfer, using request structures called "URBs" (USB Request Blocks).
Accordingly, the USB Core API exposed to device drivers covers quite a lot of territory. You'll probably need to consult the USB 2.0 specification, available online from www.usb.org at no cost, as well as class or device specifications.
The only host-side drivers that actually touch hardware (reading/writing registers, handling IRQs, and so on) are the HCDs. In theory, all HCDs provide the same functionality through the same API. In practice, that's becoming more true on the 2.5 kernels, but there are still differences that crop up especially with fault handling. Different controllers don't necessarily report the same aspects of failures, and recovery from faults (including software-induced ones like unlinking an URB) isn't yet fully consistent. Device driver authors should make a point of doing disconnect testing (while the device is active) with each different host controller driver, to make sure drivers don't have bugs of their own as well as to make sure they aren't relying on some HCD-specific behavior. (You will need external USB 1.1 and/or USB 2.0 hubs to perform all those tests.)