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Minecraft seeds are a series of characters.
This sequence of characters is folded back on itself a dozen different times and in different ways, and produces a series of iterations and variations for different chunks, different cave-patterns, different biomes....etc. The world is built algorithmically using the seed as a base. Chunks are loaded one at a time, and refer back to the seed to generate themselves. This means that returning to the same area on the same seed will always look the same, no matter what. As you explore the world, information is stored in memory. What mobs are spawned in the seed, what changes have been made to the block positions, these are stored upon leaving a chunk, and reloaded when you come near it again.
However, the seed is only so large a number. So eventually, this algorithm must terminate. That's what happens at the border of a minecraft world, the seed runs dry and begins to repeat itself. When does this happen? Each minecraft seed creates a world with more surface area than the planet Neptune. 30,000,000 blocks from the center to the edge.
Minecraft territory is, for all intents and purposes, infinite. You will never run out of diamond, or emerald, or villager, or any resource at all. It's basically infinite.
