A host controller interface (HCI) is a register level interface which allows a host controller for USB or FireWire to communicate with the operating system of a personal computer.
On the expansion card or motherboard controller, this involves much custom logic, with digital logic engines in FPGAs plus analog circuitry managing the high speed differential signals. On the software side, it requires a device driver (called a Host Controller Driver, or HCD).
Open Host Controller Interface, or OHCI, is an open standard.
When applied to a FireWire card, OHCI means that the card supports a standard interface to the PC and can be used by the OHCI FireWire drivers that come with all modern operating systems. Because the card has a standard OHCI interface, the OS does not need to know in advance exactly who makes the card or how it works, it can safely assume that the card understands the set of well-defined commands that are defined in the standard protocol.
The OHCI standard for USB is similar, but supports USB 1.1 (full and low speeds) only; so as a result its register interface looks completely different. Compared to UHCI, it moves more intelligence into the controller so is accordingly much more efficient; this was part of the motivation for defining it. If a computer provides non-x86 USB 1.1, or x86 USB 1.1 without an Intel or VIA chipset, it probably uses OHCI (e.g. OHCI is common on add-in PCI Cards based on an NEC chipset).
Far from being "universal", it is actually proprietary and is incompatible with OHCI. Intel and VIA controllers generally use UHCI, while other vendors use OHCI.
Enhanced Host Controller Interface (EHCI) is a high speed controller standard which is publicly specified. The USB-IF insisted on this for USB 2.0 instead of having a different standard for PCI-based USB interfaces, which would have increased complexity and therefore costs. Intel hosted the EHCI conformance testing, which helped to prevent divergence from the standard.
EHCI only provides high speed USB functions. It relies on a "companion controller", either OHCI or UHCI, to handle full and low speed devices. Motherboards and PCI Cards that provide high speed ports thus have two controllers, one handling high speed devices and the other handling low and full speed devices.
It is not uncommon to find UHCI, OHCI and EHCI all co-existing within a standard PC computer, with a UHCI driver providing low and full speed functions on the (Intel chipset) motherboard, an OHCI driver providing low and full speed functions for the USB ports on an add-in (NEC chipset) PCI expansion card, and an EHCI driver providing high speed functions for the USB ports on that expansion card.