Developed in house by Steinberg, and eventually released to the public with an SDK, ASIO (A-zee-oh) was the first successful low-latency native driver format. Many people in the audio biz might not like some things Steinberg does, but the fact remains they have established more than one de facto standard for the industry, and they probably could exploit that more than they do.
ASIO comes in two flavors, old school ASIO and ASIO 2. The original ASIO spec was the first to get latency down low enough that native audio programs started to look like viable ways to get results. One of the downsides of ASIO was that it did NOT support multiple applications. The ASIO application would simply take over the entire card. This allowed for tight integration with the hardware, as well as quite sophisticated routing options, but if you wanted to monitor anything else, like a separate editor for example, you had to get a second card. ASIO also had no sync options, relying on MTC just like DirectX.
ASIO 2 changed most of this for the better. First, it includes "direct monitoring" which is a way to keep record latencies very low. The easiest way to think of this is that the driver splits all of the inputs on the sound card. One set is passed to the CPU for recording and effects, the other set is sent immediately to the outputs on the sound card. In practice this means that you can play and record at the same time, the only down side is that any effects you have running in the audio app (like a reverb patch or a delay) won't be heard when you are recording. ASIO 2 also added much greater sync capabilities. For cards with hardware to support it, ASIO 2 is capable of syncing with sample accuracy to another application, and can also take transport commands (record, play, stop, and so on) from another app. Not many cards do this yet, but it is a potentially wonderful thing for desktop recording.
Written by the other participant in the Great Teutonic Audio Wars, EASI is the answer to ASIO developed by Emagic. Pronounced E-Z, it might sound like K-Tel but it's far from kitsch. Emagic openly complained about what they felt was a lack of openness from Steinberg with ASIO, and they have gone out of their way to include information about EASI for free and with no strings attached. To understand this, it is important to remember that Emagic and Steinberg are both huge, successful, powerful vendors of cross platform audio apps. If you want to start a bloody brawl similar to yelling "vi rules" at an emacs convention, go to a Logic users group and tell them how much booty Cubase kicks. You'll understand this rivalry in no time.
EASI includes much of what ASIO does. It has the same high-resolution multi-channel support, the ability to sync with sample accuracy and pass transport information. Emagic claims a few extra things beyond this. First, the SDK is available without signing any agreements with Emagic. Seems they were upset with someone making them sign something at some point. Second, they claim a very high level of platform portability, and ease of development. Lastly, EASI allows the use of onboard DSP by the EASI host application. This is a potentially killer thing for EASI. Many cards have some kind of DSP chip in addition to the converters on the sound card. Generally, they sit there unused because the host application doesn't know how to use them. Emagic built in the capability for the driver developer to use that currently dormant DSP for addition mixing, effects, or whatever.
Nobody seriously questions the ability of Emagic to write killer audio code. However, the main drawback of EASI is lack of current card support. Logic is the only host app that really uses EASI drivers, and outside of Emagic's own cards, not many have EASI drivers. Still, their technical capabilities and market share may let them play effective catch up to Steinberg's ASIO spec.