When answering this question, you have to remember the difference between a nation and a state in political science terminology. A nation is not a country -- it is a group of people who identify with one another on ethnic grounds. And a state is not a subset of a country (like Wisconsin or Washington). It is, instead, an actual whole country.
So, using these definitions, a multinational state is a country that has citizens of many different ethnic groups. These ethnic groups would see themselves as fundamentally different from other groups within their same country. The Soviet Union was a great example of this because it encompassed many different nations (Uzbeks, Kazakhs, Ukrainians, Russians, etc).
A nation-state, by contrast, is one where the country has only one ethnic group. Perhaps the clearest example of such a state today would be Japan since only ethnic Japanese are considered to truly be full citizens of that country.
The difference between a multinational-state and a nation-state is pretty basic and hinges on simple definitions of sometimes complex concepts. Let's start with the definition of "nation" for purposes of discussion. Unlike "state," which is entirely a political concept, a "nation" is more ethnographic. This means that a nation is a group of people who self-identify with each other on the basis of ethnicity, ideology, religion, language, and any other similarity that binds the parts into an identifiable whole. The United States of America is a nation because, despite its multitude of ethnicities and religions, there exists (although, not as much as in the past) a common identification with the legal underpinnings from which the country was born, in effect, the accepted legitimacy of the U.S. Constitution. The United States, historically, was unique in this regard. Most of the rest of the world, for hundreds of years, formed nations almost entirely on the basis of a common ethnicity and/or religion. The partition of India in 1947 into a Hindu-majority India and a Muslim-majority Pakistan was a case of the formation of two independent nations based on religion.
Now, let's look at the definition of a "state." As noted, "states" are entirely political constructs that may or may not reflect the dominance of a single nation of people. States are defined by their recognizable borders, irrespective of whether those borders, as in Africa and the Middle East, were drawn by outsiders, and by the consent of the populace within those borders to be governed by a central institution or set of institutions. Africa and the Middle East are noteworthy for the fact that many borders within those vast regions were drawn by British and French colonial administrators, and many of those borders were drawn with little or no regard for recognition of nations. In other words, political considerations determined the borders of independent states rather than recognition of identifiable nations. Always remember: state borders were often arbitrarily drawn and frequently bisected tribes, clans, nations, etc.
Bringing it all together, a nation-state is a state comprised of one identifiable group of people. They might be identifiable in terms of ethnicity or language, or political outlook, but they are common in at least one major sense. A multinational-state, then, is a state comprised of more than one nation, such as existed in India prior to partition (although, India remains something of a multinational-state given its very large Muslim and Sikh populations). Pakistan, the region that broke away from India to form a majority Muslim nation-state, itself experienced a major and very bloody fracture in 1971 when East Pakistan broke away and declared its independence from West Pakistan and called itself Bangladesh. Anyway, the point is, multinational-states are infinitely more complex than nation-states because of their diversity of cultures, languages, ethnicities, etc.
As a side-note, there are also what are called "city-states." City-states have their origins in ancient Greece and Italy--both cases of eventual unification of those disparate city-states into one large state. Today, one of the more prominent examples of a city-state is Singapore, which enjoys sovereignty and independence despite its tiny size, and which is overwhelmingly comprised of a common people who view themselves as distinct from Malaysia and China despite their historical and cultural linkages to both.